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Chapter 0: Problem 11

Evaluate each exponential expression. $$4^{-3}$$

Short Answer

Expert verified
The result of \(4^{-3}\) is thus \(\frac{1}{64}\).

Step by step solution

01

Understanding Negative Exponents

A number with a negative exponent is simply just the reciprocal of the number with the same positive exponent. In other words, \(a^{-n}\) is equal to \(\frac{1}{a^n}\). So, \(4^{-3}\) is equal to \(\frac{1}{4^3}\).
02

Evaluating Positive Exponential Expression

Now, you evaluate the expression in the denominator, which is now a positive exponent. This means that 4 is getting multiplied by itself 3 times. So, \(4^3 = 4*4*4 = 64\). Then you substitute back into the fraction: \(\frac{1}{4^3} = \frac{1}{64}\).
03

Simplifying the Expression

Since it's already a simplified fraction, the solution is thus \(\frac{1}{64}\).

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Key Concepts

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

Reciprocals
Understanding reciprocals can be a game-changer when you tackle negative exponents. Simply put, the reciprocal of a number is 1 divided by that number. For instance, the reciprocal of any integer like 4 is given by \( \frac{1}{4} \).

When you see a negative exponent, such as \( 4^{-3} \), it simply signals that you need to take the reciprocal of the base raised to the positive version of the exponent. The negative sign tells you to "flip" the number into its reciprocal.

This transformation makes it easier to handle problems by converting a seemingly complex expression into a fraction, setting the stage for further simplification.
Positive Exponents
Once you have turned the base with a negative exponent into its reciprocal, you will work with positive exponents. When you see an expression like \( 4^3 \), you need to think about what the exponent tells you to do.

Positive exponents indicate how many times you should multiply the base by itself. In our example, \( 4^3 \) means you multiply 4 by itself three times:
  • \( 4 \times 4 = 16 \)
  • Then \( 16 \times 4 = 64 \)

This result (64) becomes the new denominator in the reciprocal expression, making it easier to understand the overall value of the original problem.
Simplifying Fractions
After evaluating an expression with exponents, you're often left with a fraction. Simplifying fractions means ensuring there's no further division possible between the numerator and the denominator.

In the expression \( \frac{1}{64} \), the fraction is already as simple as it can be, as 1 is already the smallest positive whole number. In such cases, it's reassuring to know that your work is complete because the fraction cannot be reduced further!

Understanding when a fraction is fully simplified helps in grasping the simplicity and precision of mathematical expressions, allowing you to move confidently to other problems.

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Most popular questions from this chapter

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