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Rational exponents and surds

The laws of exponents can also be extended to include the rational numbers. A rational number is any number that can be written as a fraction with an integer in the numerator and in the denominator. We also have the following definitions for working with rational exponents.

Identity 1

  • If \(r^n = a\), then \(r = \sqrt[n]{a} \quad (n \geq 2)\)
  • \(a^{\frac{1}{n}} = \sqrt[n]{a}\)
  • \(a^{-\frac{1}{n}} = (a^{-1})^{\frac{1}{n}} = \sqrt[n]{\frac{1}{a}}\)
  • \(a^{\frac{m}{n}} = (a^{m})^{\frac{1}{n}} = \sqrt[n]{a^m}\)
where \(a > 0\), \(r > 0\) and \(m,n \in ℤ\), \(n \ne 0\).

For \(\sqrt{25} = 5\), we say that 5 is the square root of 25 and for \(\sqrt[3]{8} = 2\), we say that 2 is the cube root of 8. For \(\sqrt[5]{32} = 2\), we say that 2 is the fifth root of 32.

When dealing with exponents, a root refers to a number that is repeatedly multiplied by itself a certain number of times to get another number. A radical refers to a number written as shown below.

Image

The radical symbol and degree show which root is being determined. The radicand is the number under the radical symbol.

  • If \(n\) is an even natural number, then the radicand must be positive, otherwise the roots are not real. For example, \(\sqrt[4]{16} = 2\) since \(2 \times 2 \times 2 \times 2 = 16\), but the roots of \(\sqrt[4]{-16}\) are not real since \((-2) \times (-2) \times (-2) \times (-2) \ne -16\).

  • If \(n\) is an odd natural number, then the radicand can be positive or negative. For example, \(\sqrt[3]{27} = 3\) since \(3 \times 3 \times 3 = 27\) and we can also determine \(\sqrt[3]{-27} = -3\) since \((-3) \times (-3) \times (-3) = -27\).

It is also possible for there to be more than one n root of a number. For example, \(\left(-2\right)^2 = 4\) and \(2^2 = 4\), so both −2 and 2 are square roots of 4.

A surd is a radical which results in an irrational number. Irrational numbers are numbers that cannot be written as a fraction with the numerator and the denominator as integers. For example, \(\sqrt{12}\), \(\sqrt[3]{\textrm{100}}\), \(\sqrt[5]{25}\) are surds.

Example 1: Rational exponents

Question

Write each of the following as a radical and simplify where possible:

  1. \(18^\frac{1}{2}\)
  2. \((-\textrm{125})^{-\frac{1}{3}}\)
  3. \(4^\frac{3}{2}\)
  4. \((-81)^\frac{1}{2}\)
  5. \((\textrm{0,008})^\frac{1}{3}\)

Answer

  1. \(18^\frac{1}{2} = \sqrt{18}\)
  2. \((-\textrm{125})^{-\frac{1}{3}} = \sqrt[3]{(-\textrm{125})^{-1}} = \sqrt[3]{\dfrac{1}{-\textrm{125}}} = \sqrt[3]{\dfrac{1}{(-5)^3}} = -\dfrac{1}{5}\)
  3. \(4^\frac{3}{2} = (4^3)^{\frac{1}{2}} = \sqrt{4^3} = \sqrt{64} = 8\)
  4. \((-81)^\frac{1}{2} = \sqrt{-81} =\) not real
  5. \((\textrm{0,008})^\frac{1}{3} = \sqrt[3]{\dfrac{8}{\textrm{1000}}} = \sqrt[3]{\dfrac{2^3}{10^3}} = \frac{2}{10} = \dfrac{1}{5}\)

Example 2: Rational exponents

Question

Simplify without using a calculator:

Answer

Write the fraction with positive exponents in the denominator

\[\left( \dfrac{5}{\frac{1}{4} - \frac{1}{9}} \right)^{\frac{1}{2}}\]

Simplify the denominator

\[\begin{align} &= \left( \dfrac{5}{\frac{9-4}{36}} \right)^{\frac{1}{2}} \\ &= \left( \dfrac{5}{\frac{5}{36}} \right)^{\frac{1}{2}} \\ &= \left( 5 \div \frac{5}{36} \right)^{\frac{1}{2}} \\ &= \left( 5 \times \frac{36}{5} \right)^{\frac{1}{2}} \\ &= (36)^{\frac{1}{2}} \end{align}\]

Take the square root

\[\begin{align} &= \sqrt{36}\\ &= 6 \end{align}\]

Exercise 1: Rational exponents and surds

Simplify the following and write answers with positive exponents:

  1. \(\sqrt{49}\)

  2. \(\sqrt{36^{-1}}\)

  3. \(\sqrt[3]{6^{-2}}\)

  4. \(\sqrt[3]{-\dfrac{64}{27}}\)

  5. \(\sqrt[4]{(16x^4)^3}\)

  1. \(\sqrt{49} = 7\)

  2. \[\begin{align*} 36^{-\frac{1}{2}} &= \frac{1}{36^{\frac{1}{2}}} \\ &= \frac{1}{\sqrt{36}} \\ &= \frac{1}{6} \end{align*}\]
  3. \[\begin{align*} 6^{-\dfrac{2}{3}} &= \frac{1}{6^{\frac{2}{3}}}\\ &= \frac{1}{\sqrt[3]{6^2}} \\ &= \frac{1}{\sqrt[3]{36}} \end{align*}\]
  4. \[\begin{align*} \sqrt[3]{-\dfrac{64}{27}} &= \sqrt[3]{-\frac{4^3}{3^3}} \\ &= \sqrt[3]{ \left( -\frac{4}{3} \right)^3} \\ &= -\frac{4}{3} \end{align*}\]
  5. \[\begin{align*} \sqrt[4]{(16x^4)^3} &= \left( (16x^4)^3 \right)^{\frac{1}{4}} \\ &= \left( 2^4x^4 \right)^{\frac{3}{4}} \\ &= 2^3x^3 \\ &= 8x^3 \end{align*}\]

Simplify:

  1. \({s}^{\frac{1}{2}}÷{s}^{\frac{1}{3}}\)

  2. \({\left(64{m}^{6}\right)}^{\frac{2}{3}}\)

  3. \(\dfrac{12m^{ \frac{7}{9}}}{8m^{- \frac{11}{9}}}\)
  4. \(\left(5x\right)^0 + 5x^0 - \left(\textrm{0,25}\right)^{-\textrm{0,5}} + 8^{\frac{2}{3}}\)

  1. \[\begin{align*} s^{\frac{1}{2}} \div s^{\frac{1}{3}} &= s^{\frac{1}{2}} \times \frac{1}{s^{\frac{1}{3}}} \\ &= s^{\frac{1}{2} - \frac{1}{3}} \\ &= s^{\frac{3}{6} - \frac{2}{6}} \\ &= s^{\frac{1}{6}} \end{align*}\]
  2. \[\begin{align*} \left( 64m^6 \right)^{\frac{2}{3}} &= \left( 2^6 \right)^{\frac{2}{3}} \left( m^6 \right)^{\frac{2}{3}} \\ &= \left( 2^4 \right) \left( m^4 \right) \\ &= 16m^4 \end{align*}\]
  3. \[\begin{align*} \dfrac{12m^{\frac{7}{9}}}{8m^{-\frac{11}{9}}} &= \frac{3}{2} m^{ \frac{7}{9} - \left( - \frac{11}{9} \right)} \\ &= \frac{3}{2} m^{\frac{18}{9}} \\ &= \frac{3}{2}m^{2} \end{align*}\]
  4. \[\begin{align*} \left(5x\right)^0 + 5x^0 - \left(\textrm{0,25}\right)^{-\textrm{0,5}} + 8^{\frac{2}{3}} &= \left(1 \right) + 5(1)- \left( \frac{1}{4} \right)^{-\frac{1}{2}} + \left(2^3 \right)^{\frac{2}{3}} \\ &= 6 - 4^{\frac{1}{2}} + 4 \\ &= 10 - 2 \\ &= 8 \end{align*}\]

Use the laws to re-write the following expression as a power of \(x\):

\(x\sqrt{x\sqrt{x\sqrt{x\sqrt{x}}}}\)
\[\begin{align*} x \sqrt{x \sqrt{x\sqrt{x\sqrt{x}}}} &= x \times x^{\frac{1}{2}} \times x^{\frac{1}{4}} \times x^{\frac{1}{8}} \times x^{\frac{1}{16}} \\ &= x^{\frac{16}{16}} \times x^{\frac{8}{16}} \times x^{\frac{4}{16}} \times x^{\frac{2}{16}}\times x^{\frac{1}{16}} \\ &= x^{\frac{31}{16}} \end{align*}\]

Simplification of surds

We have seen in previous examples and exercises that rational exponents are closely related to surds. It is often useful to write a surd in exponential notation as it allows us to use the exponential laws.

The additional laws listed below make simplifying surds easier:

Identity 2

  • \(\sqrt[n]{a}\sqrt[n]{b} = \sqrt[n]{ab}\)
  • \(\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}\)
  • \(\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}\)
  • \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
  • \(\left( \sqrt[n]{a} \right)^m = a^{\frac{m}{n}}\)

Example 3: Simplifying surds

Question

Show that:

  1. \(\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab}\)
  2. \(\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}\)

Answer

  1. \[\begin{align} \sqrt[n]{a} \times \sqrt[n]{b} &= a^{\frac{1}{n}} \times b^{\frac{1}{n}} \\ &= (ab)^{\frac{1}{n}} \\ &= \sqrt[n]{ab} \end{align}\]
  2. \[\begin{align} \sqrt[n]{\frac{a}{b}} &= \left( \frac{a}{b} \right)^{\frac{1}{n}} \\ &= \dfrac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}} \\ &= \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} \end{align}\]

Examples:

  1. \(\sqrt{2} \times \sqrt{32} = \sqrt{2 \times 32} = \sqrt{64} = 8\)

  2. \(\dfrac{\sqrt[3]{24}}{\sqrt[3]{3}} = \sqrt[3]{\dfrac{24}{3}} = \sqrt[3]{8} = 2\)

  3. \(\sqrt{\sqrt{81}} = \sqrt[4]{81} = \sqrt[4]{3^4} = 3\)

Like and unlike surds

Two surds \(\sqrt[m]{a}\) and \(\sqrt[n]{b}\) are like surds if \(m=n\), otherwise they are called unlike surds. For example, \(\sqrt{\frac{1}{3}}\) and \(-\sqrt{61}\) are like surds because \(m = n = 2\). Examples of unlike surds are \(\sqrt[3]{5} \text{ and } \sqrt[5]{7y^3}\) since \(m \ne n\).

Simplest surd form

We can sometimes simplify surds by writing the radicand as a product of factors that can be further simplified using \(\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}\).

Example 4: Simplest surd form

Question

Write the following in simplest surd form: \(\sqrt{50}\)

Answer

Write the radicand as a product of prime factors

\[\begin{align} \sqrt{50} &= \sqrt{5 \times 5 \times 2} \\ &= \sqrt{5^2 \times 2} \end{align}\]

Simplify using \(\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}\)

\[\begin{align} &= \sqrt{5^2} \times \sqrt{2} \\ &= 5 \times \sqrt{2} \\ &= 5\sqrt{2} \end{align}\]

Sometimes a surd cannot be simplified. For example, \(\sqrt{6}, \sqrt[3]{30} \text{ and } \sqrt[4]{42}\) are already in their simplest form.

Example 5: Simplest surd form

Question

Write the following in simplest surd form: \(\sqrt[3]{54}\)

Answer

Write the radicand as a product of prime factors

\[\begin{align} \sqrt[3]{54} &= \sqrt[3]{3 \times 3 \times 3 \times 2} \\ &= \sqrt[3]{3^3 \times 2} \end{align}\]

Simplify using \(\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}\)

\[\begin{align} &= \sqrt[3]{3^3} \times \sqrt[3]{2} \\ &= 3 \times \sqrt[3]{2} \\ &= 3\sqrt[3]{2} \end{align}\]

Example 6: Simplest surd form

Question

Simplify: \(\sqrt{\textrm{147}}+\sqrt{\textrm{108}}\)

Answer

Write the radicands as a product of prime factors

\[\begin{align*} \sqrt{\textrm{147}} + \sqrt{\textrm{108}} &= \sqrt{49 \times 3} + \sqrt{36 \times 3} \\ &= \sqrt{7^2 \times 3} + \sqrt{6^2 \times 3} \end{align*}\]

Simplify using \(\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}\)

\[\begin{align} &= \left( \sqrt{7^2} \times \sqrt{3} \right) + \left( \sqrt{6^2} \times \sqrt{3} \right)\\ &= \left( 7 \times \sqrt{3} \right) + \left( 6 \times \sqrt{3} \right) \\ &= 7\sqrt{3} + 6\sqrt{3} \end{align}\]

Simplify and write the final answer

\[13 \sqrt{3}\]

Example 7: Simplest surd form

Question

Simplify: \(\left( \sqrt{20} - \sqrt{5} \right)^2\)

Answer

Factorise the radicands were possible

\[\left( \sqrt{20} - \sqrt{5} \right)^2 = \left( \sqrt{4 \times 5} - \sqrt{5} \right)^2\]

Simplify using \(\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}\)

\[\begin{align} &= \left( \sqrt{4} \times \sqrt{5} - \sqrt{5} \right)^2 \\ &= \left( 2 \times \sqrt{5} - \sqrt{5} \right)^2 \\ &= \left( 2\sqrt{5} - \sqrt{5} \right)^2 \end{align}\]

Simplify and write the final answer

\[\begin{align} &= \left( \sqrt{5} \right)^2 \\ &= 5 \end{align}\]

Example 8: Simplest surd form with fractions

Question

Write in simplest surd form: \(\sqrt{75} \times \sqrt[3]{(48)^{-1}}\)

Answer

Factorise the radicands were possible

\[\begin{align} \sqrt{75} \times \sqrt[3]{(48)^{-1}} &= \sqrt{25 \times 3} \times \sqrt[3]{\frac{1}{48}} \\ &= \sqrt{25 \times 3} \times \frac{1}{\sqrt[3]{8 \times 6}} \end{align}\]

Simplify using \(\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}\)

\[\begin{align} &= \sqrt{25} \times \sqrt{3} \times \frac{1}{\sqrt[3]{8} \times \sqrt[3]{6}} \\ &= 5 \times \sqrt{3} \times \frac{1}{2 \times \sqrt[3]{6}} \end{align}\]

Simplify and write the final answer

\[\begin{align} &= 5\sqrt{3} \times \frac{1}{2\sqrt[3]{6}} \\ &= \frac{5\sqrt{3}}{2\sqrt[3]{6}} \end{align}\]

Exercise 2: Simplification of surds

Simplify the following and write answers with positive exponents:

  1. \(\sqrt[3]{16} \times \sqrt[3]{4}\)

  2. \(\sqrt{a^2b^3} \times \sqrt{b^5c^4}\)

  3. \(\dfrac{\sqrt{12}}{\sqrt{3}}\)

  4. \(\sqrt{x^2y^{13}} \div \sqrt{y^5}\)

  1. \(\sqrt[3]{16} \times \sqrt[3]{4} = \sqrt[3]{16 \times 4} = \sqrt[3]{64} = 4\)

  2. \(\sqrt{a^2b^3} \times \sqrt{b^5c^4} = \sqrt{a^2b^8c^4} = ab^4c^2\)

  3. \(\dfrac{\sqrt{12}}{\sqrt{3}} = \sqrt{\dfrac{12}{3}} = \sqrt{4} = 2\)

  4. \(\sqrt{x^2y^{13}} \div \sqrt{y^5} = \sqrt{\frac{x^2y^{13}}{y^5}} = \sqrt{x^2y^8} = xy^4\)

Simplify the following:

  1. \(\left( \dfrac{1}{a} - \dfrac{1}{b} \right)^{-1}\)

  2. \(\dfrac{b-a}{a^{\frac{1}{2}} - b^{\frac{1}{2}}}\)

  1. \[\begin{align*} \left( \dfrac{1}{a} - \dfrac{1}{b} \right)^{-1} &= \left( \dfrac{b - a}{ab} \right)^{-1} \\ &= \frac{ab}{b - a} \end{align*}\]
  2. \[\begin{align*} \dfrac{b-a}{a^{\frac{1}{2}} - b^{\frac{1}{2}}} &= - \dfrac{a - b}{a^{\frac{1}{2}} - b^{\frac{1}{2}}} \\ &= - \dfrac{ \left( a^{\frac{1}{2}} \right)^2- \left( b^{\frac{1}{2}} \right)^2}{a^{\frac{1}{2}} - b^{\frac{1}{2}}} \\ &= - \dfrac{ \left( a^{\frac{1}{2}} - b^{\frac{1}{2}} \right)\left( a^{\frac{1}{2}} + b^{\frac{1}{2}} \right) } {a^{\frac{1}{2}} - b^{\frac{1}{2}}} \\ &= - \left( a^{\frac{1}{2}} + b^{\frac{1}{2}} \right) \end{align*}\]

Rationalising denominators

It is often easier to work with fractions that have rational denominators instead of surd denominators. By rationalising the denominator, we convert a fraction with a surd in the denominator to a fraction that has a rational denominator.

Example 9: Rationalising the denominator

Question

Rationalise the denominator: \[\frac{5x-16}{\sqrt{x}}\]

Answer

Multiply the fraction by \(\frac{\sqrt{x}}{\sqrt{x}}\)

Notice that \(\frac{\sqrt{x}}{\sqrt{x}}=1\), so the value of the fraction has not been changed.

\[\frac{5x - 16}{\sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{\sqrt{x}(5x - 16)}{\sqrt{x} \times \sqrt{x}}\]

Simplify the denominator

\[\begin{align} &= \frac{\sqrt{x}(5x - 16)}{\left( \sqrt{x}\right)^2} \\ &= \frac{\sqrt{x}(5x - 16)}{x} \end{align}\]

The term in the denominator has changed from a surd to a rational number. Expressing the surd in the numerator is the preferred way of writing expressions.

Example 10: Rationalising the denominator

Question

Write the following with a rational denominator: \[\frac{y-25}{\sqrt{y}+5}\]

Answer

Multiply the fraction by \(\frac{\sqrt{y}-5}{\sqrt{y}-5}\)

To eliminate the surd from the denominator, we must multiply the fraction by an expression that will result in a difference of two squares in the denominator.

\[\frac{y-25}{\sqrt{y}+5} \times \frac{\sqrt{y}-5}{\sqrt{y}-5}\]

Simplify the denominator

\[\begin{align} &= \frac{(y-25)(\sqrt{y}-5)}{(\sqrt{y}+5)(\sqrt{y}-5)} \\ &= \frac{(y-25)(\sqrt{y}-5)}{(\sqrt{y})^2 -25} \\ &= \frac{(y-25)(\sqrt{y}-5)}{y-25} \\ &= \sqrt{y}-5 \end{align}\]

Exercise 3: Rationalising the denominator

Rationalise the denominator in each of the following:

  1. \(\dfrac{10}{\sqrt{5}}\)

  2. \(\dfrac{3}{\sqrt{6}}\)

  3. \(\dfrac{2}{\sqrt{3}} \div \dfrac{\sqrt{2}}{3}\)

  4. \(\dfrac{3}{\sqrt{5}-1}\)

  5. \(\dfrac{x}{\sqrt{y}}\)

  6. \(\dfrac{\sqrt{3} + \sqrt{7}}{\sqrt{2}}\)

  7. \(\dfrac{3\sqrt{p} - 4}{\sqrt{p}}\)

  8. \(\dfrac{t-4}{\sqrt{t} + 2}\)

  9. \(\left( 1 + \sqrt{m} \right)^{-1}\)

  10. \(a \left( \sqrt{a} \div \sqrt{b} \right)^{-1}\)

  1. \[\begin{align*} \dfrac{10}{\sqrt{5}} &=\frac{10}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} \\ &=\frac{10\sqrt{5}}{5} \\ &= 2\sqrt{5} \end{align*}\]
  2. \[\begin{align*} \dfrac{3}{\sqrt{6}} &=\frac{3}{\sqrt{6}} \times \dfrac{\sqrt{6}}{\sqrt{6}} \\ &=\frac{3\sqrt{6}}{6} \\ &=\frac{\sqrt{6}}{2} \end{align*}\]
  3. \[\begin{align*} \dfrac{2}{\sqrt{3}} \div \dfrac{\sqrt{2}}{3} &=\dfrac{2}{\sqrt{3}} \div \dfrac{\sqrt{2}}{3} \\ &=\dfrac{2}{\sqrt{3}} \times \dfrac{3}{\sqrt{2}} \\ &=\dfrac{6}{\sqrt{6}} \times \dfrac{\sqrt{6}}{\sqrt{6}} \\ &=\frac{6\sqrt{6}}{6} \\ &=\sqrt{6} \end{align*}\]
  4. \[\begin{align*} \dfrac{3}{\sqrt{5}-1} &=\dfrac{3}{\sqrt{5}-1} \times \dfrac{\sqrt{5}+1}{\sqrt{5}+1} \\ &=\dfrac{3\sqrt{5}+3}{5-1} \\ &= \dfrac{3\sqrt{5}+3}{4} \end{align*}\]
  5. \[\begin{align*} \dfrac{x}{\sqrt{y}} &=\dfrac{x}{\sqrt{y}}\times \dfrac{\sqrt{y}}{\sqrt{y}} \\ &=\dfrac{x\sqrt{y}}{y} \end{align*}\]
  6. \[\begin{align*} \dfrac{\sqrt{3} + \sqrt{7}}{\sqrt{2}} &=\dfrac{\sqrt{3}+\sqrt{7}}{\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}} \\ &=\dfrac{\sqrt{3}\sqrt{2}+\sqrt{7}\sqrt{2}}{2} \\ &=\dfrac{\sqrt{6}+\sqrt{14}}{2} \end{align*}\]
  7. \[\begin{align*} \dfrac{3\sqrt{p} - 4}{\sqrt{p}} &=\dfrac{3\sqrt{p}-4}{\sqrt{p}} \times \dfrac{\sqrt{p}}{\sqrt{p}} \\ &=\dfrac{3\left ( \sqrt{p} \right )^2-4\left ( \sqrt{p} \right )}{p} \\ &=\dfrac{3p-4\sqrt{p}}{p} \end{align*}\]
  8. \[\begin{align*} \dfrac{t-4}{\sqrt{t} + 2} &=\dfrac{t-4}{\sqrt{t}+2} \times \dfrac{\sqrt{t}-2}{\sqrt{t}-2} \\ &=\dfrac{\left ( t-4 \right )\left ( \sqrt{t}-2 \right )}{t-4} \\ &=\sqrt{t}-2 \end{align*}\]
  9. \[\begin{align*} \left( 1 + \sqrt{m} \right)^{-1} &=\frac{1}{1+\sqrt{m}}\times \frac{1-\sqrt{m}}{1-\sqrt{m}} \\ &=\frac{1-\sqrt{m}}{1-m} \end{align*}\]
  10. \[\begin{align*} a \left( \sqrt{a} \div \sqrt{b} \right)^{-1} &=a\left ( \sqrt{a} \times\frac{1}{\sqrt{b}} \right )^{-1} \\ &=a\left ( \frac{\sqrt{a}}{\sqrt{b}} \right )^{-1} \\ &=a\frac{\sqrt{b}}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} \\ &=\frac{a\sqrt{ab}}{a} \\ &=\sqrt{ab} \end{align*}\]