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Chapter 4: Problem 12

Evaluate each expression. Do not use a calculator. $$-16^{3 / 2}$$

Short Answer

Expert verified
The expression evaluates to -64.

Step by step solution

01

Understand the Expression

The expression given is $$-16^{3/2}$$. This expression includes a negative sign outside the base 16, which indicates that we should apply the exponent to 16 first and then apply the negative sign.
02

Simplify the Exponentiation

The fraction \(\frac{3}{2}\) as an exponent involves two steps: first taking the square root (since the denominator is 2), and then cubing the result (since the numerator is 3).Let's start with taking the square root of 16: \( \sqrt{16} = 4. \)
03

Apply the Cubing

Next, cube the result from Step 2, which is 4: \( 4^3 = 4 \times 4 \times 4 = 64. \)
04

Apply the Negative Sign

Finally, apply the negative sign outside the entire power operation: \(-16^{3/2} = -64.\)

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Key Concepts

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

Exponentiation
Exponentiation is a way to express repeated multiplication of the same number. When you see an expression like \(16^{3/2}\), it tells us to take 16 and raise it to the power of \(\frac{3}{2}\). Exponents can be tricky because they can be fractions or negative.
When you encounter a fractional exponent, like \(\frac{3}{2}\), it means you should first take a root and then raise the result to the power.
  • The denominator (2) indicates the root, which is the square root in this case.
  • The numerator (3) is the power to which you raise the result.
A fractional exponent combines rooting and raising to a power into a single step.
Square Root
A square root is a number that, when multiplied by itself, gives the original number. It鈥檚 denoted by the radical symbol \(\sqrt{}\).
  • For example, the square root of 16 is written as \(\sqrt{16}\).
  • This equals 4 because \(4 \times 4 = 16\).
Square roots are essential when dealing with fractional exponents. In the expression \(16^{3/2}\), the square root is used as the first step (the denominator of the fraction) to simplify the base of 16 before applying any further operations. So, \(\sqrt{16} = 4\).
Negative Numbers
Negative numbers are values less than zero, represented with a minus (-) sign. When they appear outside an expression, it affects the entire result. In our problem, we have \(-16^{3/2}\).
Here, the negative sign is applied after all other operations on the expression. So, you complete the exponentiation process entirely, first determining what \(16^{3/2}\) resolves to, which is 64, and then apply the negative sign:
  • This results in \(-64\).
This distinction is important because applying the negative sign incorrectly can lead to wrong results.
Simplification
Simplification is the process of breaking down an expression into a more manageable or reduced form. In math, this means performing operations such as exponentiation, rooting, and applying signs in a structured manner. Consider these simplified steps:
  • First, process any roots in an expression like finding \(\sqrt{16} = 4\).
  • Then, handle the remaining operations such as \(4^3\), which equals 64.
  • Finally, incorporate any negative signs as needed, giving us \(-64\).
By following these steps, you ensure accuracy and make the expression much easier to understand.

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