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Chapter 5: Problem 77

Evaluate using a calculator. $$ -8^{4} $$

Short Answer

Expert verified
-4096

Step by step solution

01

Understand the Expression

The given expression is \(-8^4\). It means \(8\) raised to the power of \(4\) with a negative sign in front of it.
02

Calculate the Power

First, calculate \(8^4\). Using a calculator, \(8^4 = 4096\).
03

Apply the Negative Sign

Now, apply the negative sign to the result obtained. So, \( -8^4 = -4096\).

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Key Concepts

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

Exponents
When dealing with exponents, you're working with numbers raised to a certain power. For instance, in the term \(8^4\), 8 is the base and 4 is the exponent. This means you multiply 8 by itself 4 times: \[ 8 \times 8 \times 8 \times 8 = 4096 \]. Understanding exponents is crucial in various areas of math. They show how many times to use the base in a multiplication. Note: Parentheses are important. \(-8^4\) is different from \((-8)^4\). The first means \(-(8^4)\) and is negative, whereas the second means \((-8) \times (-8) \times (-8) \times (-8) = 4096\) which is positive.
Use of Calculators in Math
Calculators are handy tools to simplify complex calculations. When evaluating expressions like \(-8^4\), calculators help ensure accuracy and save time. To calculate \(8^4\) on your calculator, type 8, press the exponent key (usually labeled as ^ or \(\texttt{EXP} \)), then type 4. Note how the calculator applies the negative sign. Enter the exponent without the negative sign first. After you get your result, apply the negative manually or use the plus/minus button if your calculator has one.
Negative Numbers
Understanding negative numbers is key in math. Negative numbers are numbers less than zero, represented with a minus sign (-). In our problem, the negative sign in \(-8^4\) modifies the result of \(8^4\). Essentially, you calculate \(8^4 = 4096\), then apply the negative, resulting in \(-4096\). Remember: Negative signs change the direction on the number line and combine differently with exponents. Negative bases with even exponents turn positive, while negative bases with odd exponents stay negative.

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Most popular questions from this chapter

Computer spreadsheet applications allow values for cells in a spreadsheet to be calculated from values in other cells. For example, if the cell Cl contains the formula $$ =\mathrm{A} 1+2 * \mathrm{B} 1 $$ the value in Cl will be the sum of the value in Al and twice the value in B1. This formula is a polynomial in the two variables Al and B1. The cell D4 contains the formula $$ =2 * \mathrm{A} 4+3 * \mathrm{B} 4 $$ What is the value in \(\mathrm{D} 4\) if the value in \(\mathrm{A} 4\) is 5 and the value in \(\mathrm{B} 4\) is \(10^{2}?\)

Is it easier to evaluate a polynomial before or after like terms have been combined? Why?

Computer spreadsheet applications allow values for cells in a spreadsheet to be calculated from values in other cells. For example, if the cell Cl contains the formula $$ =\mathrm{A} 1+2 * \mathrm{B} 1 $$ the value in Cl will be the sum of the value in Al and twice the value in B1. This formula is a polynomial in the two variables Al and B1. The cell D6 contains the formula $$ =\mathrm{Al}-0.2 * \mathrm{B} 1+0.3^{*} \mathrm{Cl} $$ What is the value in \(\mathrm{D} 6\) if the value in \(\mathrm{Al}\) is \(10,\) the value in \(\mathrm{B} 1\) is \(-3,\) and the value in \(\mathrm{Cl}\) is \(30 ?\)

Use the fact that \(10^{3} \approx 2^{10}\) to estimate each of the following powers of \(2 .\) Then compute the power of 2 with a calculator and find the difference between the exact value and the approximation. $$ 2^{31} $$

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{a^{4}}{b^{3}}\right)^{5} $$
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