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Chapter 0: Problem 15

Evaluate each exponential expression. $$\left(2^{2}\right)^{3}$$

Short Answer

Expert verified
The calculated result for the expression \((2^2)^3\) is 64.

Step by step solution

01

Simplify Inside Parentheses

Solve the exponentiation inside the parentheses first, according to rules of precedence (PEMDAS/BODMAS). Hence, calculate \(2^2 = 4\).
02

Apply Power of Power Rule

Now, apply the result of the first operation (from step 1) to the next exponent. That is, calculate \((2^2)^3 = 4^3 = 64\).

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation involving two numbers: a base and an exponent. The base is the number you're going to multiply. The exponent tells you how many times to multiply the base by itself. For example, in the expression \(2^3\), the number 2 is the base and 3 is the exponent. This means you multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\).

Using exponents can make it much simpler to handle large numbers because it provides a compact way to represent repeated multiplication. As in the exercise provided, where we have \((2^2)^3\), exponentiation allows us to simplify this complex-looking expression step-by-step using the rules of exponents.
Powers of Numbers
Powers of numbers refer to the result of raising a base to an exponent. It has a specific role in mathematics to help express numbers in a more manageable way. The power measures how many times you apply the base numbers in multiplications.

In the given exercise, initially we solve \(2^2\). Here 2 is the base and 2 is the exponent, resulting in \(2 \times 2 = 4\). Then, this result is raised to yet another power, \(3\), which results in \(4^3\). This illustrates using a power of a power, resulting in \(4 \times 4 \times 4 = 64\).

Applying the power of numbers involves using straightforward multiplication but requires attention to the rules for efficiency and accuracy.
Order of Operations
The order of operations is crucial when solving mathematical expressions. It ensures that everyone solves the problem in the same way and gets the same result. A common acronym that helps to remember the correct sequence of operations is PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

In our exercise, the parentheses tell us to solve \(2^2\) before addressing any other operations. Once \(2^2\) has been solved, yielding 4, we must then apply the power of 3, leading us to calculate \(4^3\).

Following the order of operations properly is key in ensuring that even complex expressions are solved correctly and efficiently.

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Most popular questions from this chapter

If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be heads. It can be demonstrated that a coin is unfair if \(h,\) the number of outcomes that result in heads, satisfies \(\left|\frac{h-50}{5}\right| \geq 1.645 .\) Describe the number of outcomes that determine an unfair coin that is tossed 100 times.

This will help you prepare for the material covered in the next section. Simplify and express the answer in descending powers of \(x\) : $$2 x\left(x^{2}+4 x+5\right)+3\left(x^{2}+4 x+5\right)$$

In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and \(2012 .\) Also shown is the percentage of households in which a person of faith is married to someone with no religion. GRAPH CAN'T COPY. The formula $$I=\frac{1}{4} x+26$$ models the percentage of U.S. households with an interfaith marriage, \(I, x\) years after \(1988 .\) The formula $$N=\frac{1}{4} x+6$$ models the percentage of U.S. households in which a person of faith is married to someone with no religion, \(N, x\) years after \(1988 .\) Use these models to solve Exercises \(107-108\). a. In which years will more than \(33 \%\) of U.S. households have an interfaith marriage? b. In which years will more than \(14 \%\) of U.S. households have a person of faith married to someone with no religion? c. Based on your answers to parts (a) and (b), in which years will more than \(33 \%\) of households have an interfaith marriage and more than \(14 \%\) have a faith/no religion marriage? d. Based on your answers to parts (a) and (b), in which years will more than \(33 \%\) of households have an interfaith marriage or more than \(14 \%\) have a faith/no religion marriage?

A company wants to increase the \(10 \%\) peroxide content of its product by adding pure peroxide (100\% peroxide). If \(x\) liters of pure peroxide are added to 500 liters of its \(10 \%\) solution, the concentration, \(C,\) of the new mixture is given by $$C=\frac{x+0.1(500)}{x+500}$$ How many liters of pure peroxide should be added to produce a new product that is \(28 \%\) peroxide?

Using an example, explain how to factor out the greatest common factor of a polynomial.
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