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Chapter 5: Problem 19

For Problems \(1-30\), evaluate each numerical expression. $$ -4^{\frac{5}{2}} $$

Short Answer

Expert verified
The value of \(-4^{\frac{5}{2}}\) is \(-32\).

Step by step solution

01

Understand the Expression

The given expression is \(-4^{\frac{5}{2}}\). It's important to note that this expression is equivalent to \(-1 \times (4^{\frac{5}{2}})\). This is because the negative sign is not included inside the power.
02

Simplify the Exponent

First, calculate the power of 4. The exponent is a fractional power, \(\frac{5}{2}\). This can be expressed as taking the square root of 4 and then raising it to the fifth power.
03

Calculate the Square Root

Find the square root of 4, which is \(\sqrt{4} = 2\).
04

Raise to the Power

Now, raise the result from the square root to the fifth power: \(2^5 = 32\).
05

Apply the Negative Sign

Multiply the result by -1 as per the original expression arrangement: \(-1 \times 32 = -32\).

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Key Concepts

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

Fractional Exponents
Fractional exponents are a different way to express roots and powers in mathematics. They tell us to take both a root and a power of a number and are written as fractions.
A fractional exponent like \(\frac{5}{2}\) means we perform two operations:
  • Take the square root (because the denominator is 2).
  • Raise the result to the power of 5 (because the numerator is 5).
For example, to solve \(4^{\frac{5}{2}}\), we follow these steps:
  • First, calculate the square root of 4: \(\sqrt{4} = 2\).
  • Then, raise this result to the fifth power: \(2^5 = 32\).
This process demonstrates how fractional exponents effectively combine roots and powers.
Order of Operations
The order of operations is a set of rules in mathematics that determines how to evaluate expressions properly. It's crucial to maintain consistency and avoid confusion.
The acronym PEMDAS helps remember the order:
  • P: Parentheses first.
  • E: Exponents (including fractional exponents) come next.
  • MD: Multiplication and Division (from left to right).
  • AS: Addition and Subtraction (from left to right).
In the exercise \(-4^{\frac{5}{2}}\), it is essential to address the exponent first as dictated by PEMDAS. So, solve \(4^{\frac{5}{2}}\) before applying the negative sign. This routine ensures correct results every time.
Negative Numbers
Negative numbers can change the sign of a value and are indicated by a minus sign \((-1)\).
In the expression \(-4^{\frac{5}{2}}\), the negative sign is separate from the exponent.
This implies:
  • First, evaluate the base and exponent (ignoring the negative sign for a moment).
  • The operation \(4^{\frac{5}{2}}\) results in \(32\).
  • Finally, apply the negative sign, multiplying by \((-1)\): \((-1) \times 32 = -32\).
Understanding how negative numbers interact with exponents and other operations helps avoid mistakes and ensure clarity in expressions.

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Most popular questions from this chapter

Perform the indicated operations and express answers in simplest radical form. $$ \frac{\sqrt[4]{27}}{\sqrt{3}} $$

Convert each number to scientific notation and perform the indicated operations. Express the result in ordinary decimal notation. $$ \frac{(60,000)(0.006)}{(0.0009)(400)} $$

For Problems \(1-18\), write each of the following in scientific notation. 4290

Sometimes it is more convenient to express a number as a product of a power of 10 and a number that is not between 1 and 10 . For example, suppose that we want to calculate \(\sqrt{640,000}\). We can proceed as follows: \(\sqrt{640,000}=\sqrt{(64)\left(10^{4}\right)}\) $$ \begin{aligned} &=\left((64)\left(10^{4}\right)\right)^{\frac{1}{2}} \\ &=(64)^{\frac{1}{2}}\left(10^{4}\right)^{\frac{1}{2}} \\ &=(8)\left(10^{2}\right) \\ &=8(100)=800 \end{aligned} $$ Compute each of the following without a calculator, and then use a calculator to check your answers. (a) \(\sqrt{49,000,000}\) (b) \(\sqrt{0.0025}\) (c) \(\sqrt{14,400}\) (d) \(\sqrt{0.000121}\) (e) \(\sqrt[3]{27,000}\) (f) \(\sqrt[3]{0.000064}\)

For Problems \(1-18\), write each of the following in scientific notation. 812,000
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