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Chapter 1: Problem 45

Evaluate each expression without using a calculator. $$ \left(\frac{25}{16}\right)^{-3 / 2} $$

Short Answer

Expert verified
The expression evaluates to \( \frac{64}{125} \).

Step by step solution

01

Understand Negative Exponent

The expression \( \left( \frac{25}{16} \right)^{-3/2} \) has a negative exponent. When an expression has a negative exponent, we take the reciprocal of the base and then apply the exponent as positive. Therefore, it becomes \( \left( \frac{16}{25} \right)^{3/2} \).
02

Simplify Using Fractional Exponent

The exponent \( \frac{3}{2} \) means we first take the square root, and then cube the result. Apply this to \( \frac{16}{25} \):\[ \left( \frac{16}{25} \right)^{1/2} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} \]
03

Raise to Power of 3

Now we take the result from Step 2 and raise it to the power of 3: \[ \left( \frac{4}{5} \right)^3 = \frac{4^3}{5^3} = \frac{64}{125} \]
04

Conclusion: Evaluate the Final Expression

The expression \( \left( \frac{25}{16} \right)^{-3/2} \) simplifies to \( \frac{64}{125} \).

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Key Concepts

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

Fractional Exponents
Fractional exponents might seem daunting at first, but they offer a more streamlined way to express roots and powers. Think of them as a flexible, shorthand notation.
For instance, when you see an exponent like \( \frac{3}{2} \), this can be broken down into two parts:
  • The denominator (2) indicates taking the square root.
  • The numerator (3) suggests taking that result to the third power.
Using fractional exponents means you can represent complex operations succinctly. This approach is effective for simplifying expressions involving roots. The expression \( \left( \frac{16}{25} \right)^{3/2} \) demonstrates this perfectly:
  • First, find the square root, \( \left( \frac{16}{25} \right)^{1/2} = \frac{4}{5} \).
  • Then raise this to the power of 3 to achieve the final result.
Simplifying Expressions
Simplifying expressions is like decluttering your room, making it more orderly and easier to understand. The main goal is to express the given problem in its simplest form.
Starting with the expression \( \left( \frac{16}{25} \right)^{3/2} \), simplification involves breaking it down into manageable steps:
  • First, handle the fraction by calculating the square root: \( \frac{4}{5} \).
  • Next, raise the resulting fraction to the power of 3: \( \left( \frac{4}{5} \right)^3 = \frac{64}{125} \).
Simplification doesn鈥檛 alter the value; it just makes the expression easier to read and interpret. Breaking down steps ensures clarity and reduces the risk of errors.
Evaluating Mathematical Expressions
Evaluating mathematical expressions, especially with exponents, requires a systematic approach. It鈥檚 not just about finding the answer but understanding how to arrive there.
Start with identifying any properties of exponents involved, like negative signs or fractional values. For our expression \( \left( \frac{25}{16} \right)^{-3/2} \):
  • Notice the negative exponent, which indicates reciprocal transformation.
  • Simplify the transformation by flipping the fraction: \( \left( \frac{16}{25} \right)^{3/2} \).
After this adjustment, apply the rules of fractional exponents and simplify step by step until you achieve the final evaluated value of \( \frac{64}{125} \).
Evaluating these expressions correctly enhances your mathematical intuition and prepares you for tackling even more complex problems in the future.

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Most popular questions from this chapter

Which of the following is not a polynomial, and why? $$ x^{2}+\sqrt{2} \quad x^{\sqrt{2}}+1 \quad \sqrt{2} x^{2}+1 $$

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=\sqrt{x}-1 ; g(x)=x^{3}-x^{2} $$

For each equation, find the slope \(m\) and \(y\) intercept \((0, b)\) (when they exist) and draw the graph. \(y=-3\)

Find the slope (if it is defined) of the line determined by each pair of points. \((2,3)\) and \((4,-1)\)

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=x^{3}+x ; g(x)=\frac{x^{4}+1}{x^{4}-1} $$
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