/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 \(\frac{4^{-3}}{5^{-2}}\)... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 63

\(\frac{4^{-3}}{5^{-2}}\)

Short Answer

Expert verified
The simplified form is \( \frac{25}{64} \).

Step by step solution

01

Simplify the Exponents

To simplify the given expression \( \frac{4^{-3}}{5^{-2}} \), first recognize that \( a^{-n} = \frac{1}{a^n} \). Thus, \( 4^{-3} = \frac{1}{4^3} \) and \( 5^{-2} = \frac{1}{5^2} \).
02

Rewrite the Fraction

Express \( \frac{4^{-3}}{5^{-2}} \) in terms of positive exponents: \( \frac{\frac{1}{4^3}}{\frac{1}{5^2}} \).
03

Simplify the Complex Fraction

Simplify the complex fraction by multiplying the numerator and the denominator by \( 5^2 \): \( \frac{\frac{1}{4^3} \cdot 5^2}{1} = \frac{5^2}{4^3} \).
04

Evaluate the Exponentials

Calculate \( 5^2 = 25 \) and \( 4^3 = 64 \), so the expression becomes \( \frac{25}{64} \).

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Key Concepts

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

Negative Exponents
Negative exponents might seem tricky, but they follow a simple rule. When you see a term like 4^{-3}, remember that a negative exponent means taking the reciprocal of the base raised to the positive exponent. For instance, 4^{-3} translates to \( \frac{1}{4^3} \). This is because \( a^{-n} = \frac{1}{a^n} \).

So, for any negative exponent, just flip the base to the denominator (or numerator, if it was already in the denominator) to get a positive exponent. This makes it easier to handle and simplifies calculations. The important part is to convert all negative exponents to positive ones before proceeding to further steps.
Complex Fractions
Complex fractions are fractions within fractions. They can look confusing, but breaking them down step by step makes them easier to solve. In our exercise, we had \( \frac{\frac{1}{4^3}}{\frac{1}{5^2}} \). To simplify this, you need to multiply the numerator and the denominator by the same value to clear the inner fraction.

By multiplying the numerator and the denominator by \( 5^2 \), you neutralize the inner denominator: \( \frac{\frac{1}{4^3} \cdot 5^2}{1} \). This clears the inner fraction and simplifies it to a single fraction, \( \frac{5^2}{4^3} \). Breaking down complex fractions into simpler terms always helps in solving them.
Evaluating Exponentials
Evaluating exponentials is simply calculating the value of the base raised to an exponent. In our problem, after simplifying negative exponents and complex fractions, we had \( \frac{5^2}{4^3} \). Now, let's evaluate each exponent separately: \( 5^2 = 25 \) and \( 4^3 = 64 \).

This makes our final simplification straightforward. We now have the fraction \( \frac{25}{64} \), which is the evaluated form of our original expression. Remember to take it step by step: first handle exponents, then simplify fractions, and finally calculate the values.

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Most popular questions from this chapter

Perform each indicated operation. \(\left[\left(9 b^{3}-4 b^{2}+3 b+2\right)-\left(-2 b^{3}-3 b^{2}+b\right)\right]-\left(8 b^{3}+6 b+4\right)\)

Perform each indicated operation. \(\left(4 m^{2}-3 m+2\right)+\left(5 m^{2}+13 m-4\right)+\left(-16 m^{2}-4 m+3\right)\)

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications. Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 101 \times 99 $$

Find each product. $$ -8 r^{3}\left(5 r^{2}+2\right)\left(5 r^{2}-2\right) $$

Use scientific notation to calculate the result in each expression. Write answers in scientific notation. 74\. \(\frac{3.400,000,000(0.000075)}{0.00025}\)
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