Question

Simplify each expression. $$\left(\frac{9}{8}\right)^{3 / 2}$$

Short answer

27

Step-by-step solution

Rewrite the exponent

Rewrite the expression \(\left(\frac{9}{8}\right)^{3 / 2}\) using the properties of exponents. \(\left(\frac{9}{8}\right)^{3 / 2} = \left(\frac{9}{8}\right)^{1.5}\)

Rewrite using fractional exponents

Rewrite \(\left(\frac{9}{8}\right)^{1.5}\). The fractional exponent can be expressed as the product of an integer exponent and a square root. \(\left(\frac{9}{8}\right)^{1.5} = \left(\left(\frac{9}{8}\right)^{1 / 2}\right)^3\)

Calculate the square root

Calculate the square root of \(\frac{9}{8}\). \(\left(\frac{9}{8}\right)^{1 / 2}\) is equal to \(\frac{\sqrt{9}}{\sqrt{8}} = \frac{3}{\sqrt{8}}\)

Simplify the square root

Simplify \(\frac{3}{\sqrt{8}}\) by rationalizing the denominator. Multiply both the numerator and the denominator by \(\sqrt{8}\): \(\frac{3 \sqrt{8}}{8}\)

Cube the simplified expression

Raise the simplified fraction to the power of 3: \(\left(\frac{3 \sqrt{8}}{8}\right)^3\). Simplifying further: \(\left(3 \sqrt{8}\right)^3 = 27 \sqrt{8}^3 = 27 \cdot 8 \sqrt{8} = 216 \sqrt{8}\)

Final simplified form

Combine the results to obtain the final simplified expression: \(\left(\frac{216}{8}\right) = 27\)

Key concepts

Fractional Exponents

Fractional exponents are a way to express powers and roots together, making complex expressions more manageable. For example, instead of writing \(\frac{\root{3}{27}}{\root{2}{4}}\), you can use fractional exponents: \((27^{1/3})(4^{-1/2})\). This format helps to simplify multiplication and division calculations because of the consistent rules of exponents. Remember, the numerator of a fractional exponent represents the power, and the denominator represents the root. So, \((a^{m/n})\) means \((\root{n}{a^m})\). It is crucial for transforming and simplifying expressions, such as \(\frac{(9/8)^{3/2}}=(\frac{9}{8})^{1.5}\), allowing us to handle more straightforward exponentiation. Simplifying and transforming the expressions using fractional exponents will make further simplifications and calculations easier.

Properties of Exponents

The properties of exponents are essential tools to simplify and solve expressions involving exponents efficiently. Here are some key properties:

• Product of Powers: When multiplying similar bases, add the exponents: \(a^m \times a^n = a^{m+n}\)


• Quotient of Powers: When dividing similar bases, subtract the exponents: \(a^m / a^n = a^{m-n}\)


• Power of a Power: Multiply the exponents: \( (a^m)^n = a^{mn}\)


• Power of a Product: Distribute the exponent: \( (ab)^m = a^m b^m \)


• Power of a Quotient: Distribute the exponent: \( (\frac{a}{b})^m = \frac{a^m}{b^m} \)

Using these properties, you can rewrite complex expressions in more manageable forms, making calculations easier. For example, in our expression \(\frac{9}{8}^{3/2}\), we can simplify it step-by-step using the above properties, leading us to a simplified and more understandable form.

Rationalizing the Denominator

Rationalizing the denominator is the process of eliminating any irrational numbers from the denominator of a fraction. This is done to simplify the expression and make it easier to interpret or further simplify. If you have a radical in the denominator, multiply the numerator and the denominator by a conjugate or an appropriate value to remove the radical. For example, \( \frac{3}{\root{8}}\) can be rationalized by multiplying both numerator and denominator by \( \root{8} \), resulting in \( \frac{3 \root{8}}{8} \). This process ensures that we end up with an expression with a rational number in the denominator. In our exercise, after obtaining \( \frac{3}{\root{8}} \), rationalizing yields \( \frac{3 \root{8}}{8} \), enabling us to proceed further with our simplification steps until we reach the final simplified form.