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Chapter 1: Problem 9

Evaluate each expression without using a calculator. $$ 4^{-2} \cdot 2^{-1} $$

Short Answer

Expert verified
The expression evaluates to \(\frac{1}{32}\).

Step by step solution

01

Simplify the First Exponent

The expression includes the term \(4^{-2}\). Recall that negative exponents indicate the reciprocal. Therefore, \(4^{-2}\) can be rewritten as \(\frac{1}{4^2}\). Evaluating \(4^2\) gives \(16\), so \(4^{-2} = \frac{1}{16}\).
02

Simplify the Second Exponent

The expression also includes \(2^{-1}\). Similarly, a negative exponent here indicates the reciprocal. Hence, \(2^{-1} = \frac{1}{2}\).
03

Multiply Simplified Terms

Now, multiply the simplified fractions from the previous steps: \(\frac{1}{16} \cdot \frac{1}{2}\). To multiply fractions, multiply their numerators and denominators: \(\frac{1 \cdot 1}{16 \cdot 2} = \frac{1}{32}\).
04

Conclusion

We have fully simplified the original expression. The result of \(4^{-2} \cdot 2^{-1}\) is \(\frac{1}{32}\).

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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 look intimidating at first, but they simply represent the reciprocal of the base raised to the opposite positive power. So when you see something like \( a^{-n} \), what this actually means is you'll take \( 1 \) and divide it by \( a^n \). In today's problem, each base number with a negative exponent means turning it into a fraction like this:
  • \( 4^{-2} \) becomes \( \frac{1}{4^2} \).
  • \( 2^{-1} \) becomes \( \frac{1}{2} \).
Instead of pushing numbers around, just turn the exponent negative into a denominator and the math becomes a lot easier! Remember, this trick is not only useful for numbers like 4 and 2 but for any base number.
Fraction Multiplication
Fraction multiplication might sound fancy, but it actually follows some simple steps. When you need to multiply two fractions like \( \frac{a}{b} \) and \( \frac{c}{d} \), you just need to remember to multiply straight across the fractions:
  • Multiply the numerators (the top numbers): \( a \cdot c \).
  • Multiply the denominators (the bottom numbers): \( b \cdot d \).
So, for our problem:\[ \frac{1}{16} \times \frac{1}{2} = \frac{1 \times 1}{16 \times 2} = \frac{1}{32} \]This straightforward method helps keep things less confusing. Just remember not to worry about finding a common denominator, as that's needed only for addition and subtraction of fractions.
Simplifying Expressions
Simplifying expressions involves breaking down a problem into its simplest form. Here, the process began by dealing with individual negative exponents first. By transforming these exponents into fractions, each number was easier to manage. After converting, the fractions were multiplied as described in fraction multiplication. Finally, it resulted in simplified terms that were combined into a single expression. Simplifying expressions reduces complexity and can turn a scary-looking problem into an easy solution. Always break down each step to unravel the expression one layer at a time; it's like peeling an onion, making sure you do it gently and thoroughly.

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Most popular questions from this chapter

For the quadratic function \(f(x)=a x^{2}+b x+c\), what condition on one of the coefficients will guarantee that the function has a highest value? A lowest value?

Use your graphing calculator to graph the following four equations simultaneously on the window \([-10,10]\) by \([-10,10]:\) $$ \begin{array}{l} y_{1}=3 x+4 \\ y_{2}=1 x+4 \end{array} $$ \(y_{3}=-1 x+4\) (Use \((-)\) to get \(-1 x\). \()\) $$ y_{4}=-3 x+4 $$ a. What do the lines have in common and how do they differ? b. Write the equation of a line through this \(y\) -intercept with slope \(\frac{1}{2}\). Then check your answer by graphing it with the others.

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BIOMEDICAL: Cell Growth One leukemic cell in an otherwise healthy mouse will divide into two cells every 12 hours, so that after \(x\) days the number of leukemic cells will be \(f(x)=4^{x}\). a. Find the approximate number of leukemic cells after 10 days. b. If the mouse will die when its body has a billion leukemic cells, will it survive beyond day \(15 ?\)
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