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Chapter 0: Problem 32

Use the laws of exponents to compute the numbers. $$\left(9^{4 / 5}\right)^{5 / 8}$$

Short Answer

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3

Step by step solution

01

Identify the Given Expression

The expression given is \(\big(9^{\frac{4}{5}}\big)^{\frac{5}{8}}\).
02

Apply the Power of a Power Rule

According to the power of a power rule, \( (a^m)^n = a^{m \times n} \). Here, \(a = 9\), \(m = \frac{4}{5}\) and \(n = \frac{5}{8}\).
03

Multiply the Exponents

Calculate the product of the exponents: \(\frac{4}{5} \times \frac{5}{8} = \frac{4 \times 5}{5 \times 8} = \frac{20}{40} = \frac{1}{2}\).
04

Simplify the Expression

Thus, \((9^{\frac{4}{5}})^{\frac{5}{8}}\) simplifies to \({9^{\frac{1}{2}}} = \big(9^{0.5}\big) = \big(\frac{\text{square root of } 9\big)\big} = 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

power of a power rule
When you have an expression like \((9^{\frac{4}{5}})^{\frac{5}{8}}\), we need to apply the power of a power rule to simplify it.
This rule states that \( (a^m)^n = a^{m \times n}\).
It allows us to combine the exponents by multiplying them together.
For instance, in our initial expression, we have \(a = 9\), \(m = \frac{4}{5}\), and \(n = \frac{5}{8}\).
By multiplying \(\frac{4}{5}\) by \(\frac{5}{8}\), we can make the exponents simpler. This is a powerful technique to streamline complex exponent expressions.
Always remember this rule when dealing with multiple layers of exponents. It makes solving these problems much easier.
simplification of exponents
Simplification of exponents is critical in solving problems efficiently.
In the context of our example, the simplified form \(9^{\frac{1}{2}} = \sqrt{9}\) simplifies to 3.
This simplification involves recognizing common powers and roots.
For example, \(a^{0.5}\) is equivalent to \( a^{1/2} \), the square root of \(a\).
So, \(9^{0.5} = \sqrt{9} = 3\).
Being comfortable with converting these forms is essential. It helps in quickly finding the simplest and most intuitive solution.
Practice simplifying various exponents, as it greatly enhances problem-solving efficiency.
exponent multiplication
Exponent multiplication is a straightforward but powerful tool.
It involves multiplying the numerators and denominators of fractional exponents.
In our example, this means multiplying \(\frac{4}{5}\) by \(\frac{5}{8}\), resulting in \(\frac{20}{40}\) or \(\frac{1}{2}\).
You can always simplify by canceling out common factors. Here, 5 and 4 cancel out, leaving us with 1 and 2.
This step-by-step multiplication ensures that the exponent combination remains accurate.
Understanding this concept deeply can help you tackle more complex math problems with ease.

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Most popular questions from this chapter

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\frac{\left(-27 x^{5}\right)^{2 / 3}}{\sqrt[3]{x}}$$

Use your graphing calculator to find the value of the given function at the indicated values of \(x .\) $$f(x)=x^{4}+2 x^{3}+x-5 ; \quad x=-\frac{1}{2}, x=3$$

If \(g(t)=t^{3}+5,\) find \(\frac{g(t+h)-g(t)}{h}\) and simplify.

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\frac{x^{4} \cdot y^{5}}{x y^{2}}$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\frac{x^{-4}}{x^{3}}$$
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