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Chapter 0: Problem 74

Simplify. $$16^{5 / 2}$$

Short Answer

Expert verified
1024

Step by step solution

01

- Understand the Exponent

The exponent is given as a fraction: \( \frac{5}{2} \). This means we first find the square root (denominator) and then raise it to the power of 5 (numerator).
02

- Square Root of 16

Calculate the square root of 16. \( \sqrt{16} = 4 \). So, \( 16^{\frac{1}{2}} = 4 \).
03

- Raise to the Power of 5

Now, raise 4 to the power of 5: \( 4^5 = 1024 \).

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Key Concepts

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

fractional exponents
Fractional exponents often look confusing at first, but they are a very useful way to express roots and powers. A fractional exponent is indicated by a fraction, where the numerator (top number) represents the power, and the denominator (bottom number) represents the root. For example, in the term \(a^{m/n}\), \(a\) is the base, \(m\) is the power, and \(n\) is the root. This means that \(a^{m/n}\) can be written as \((a^{1/n})^m = (\sqrt[n]{a})^m\).

Let's apply this to our problem: \(16^{5/2}\). Here, 16 is the base, 5 is the power, and 2 is the root (square root). So, first, we find the square root of 16, and then we raise that result to the power of 5.
square roots
The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of 16 is written as \(\sqrt{16}\) or \(16^{1/2}\). Simplifying this gives us positive 4, because \(4 \times 4 = 16\).

Therefore, \(\sqrt{16} = 4\).

To continue with our problem, we now know that \(16^{1/2} = 4\). Next, we move on to raising this result to the power of 5.
power rules
Power rules are the rules we apply when working with exponents. One helpful rule is that when raising a power to another power, we multiply the exponents.\((a^m)^n = a^{m \times n}\).

In our example, after calculating the square root, we need to raise the result (4) to the power of 5, written as \(4^5\).

Calculating this, \(4 \times 4 = 16\), \(16 \times 4 = 64\), \(64 \times 4 = 256\), and \(256 \times 4 = 1024\).

So, \(4^5 = 1024\).

Thus, \(16^{5/2} = 1024\). This shows how fractional exponents, square roots, and power rules come together to simplify this mathematical expression.

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Most popular questions from this chapter

Impulses in nerve fibers travel at a speed of \(293 \mathrm{ft} / \mathrm{sec}\). The distance \(D,\) in feet, traveled in \(t\) sec is given by \(D=293 t .\) How long would it take an impulse to travel from the brain to the toes of a person who is \(6 \mathrm{ft}\) tall?

Jimmy decides to mow lawns to earn money. The initial cost of his lawnmower is 250 dollars. Gasoline and maintenance costs are 4 dollars per lawn. a) Formulate a function \(C(x)\) for the total cost of mowing \(x\) lawns. b) Jimmy determines that the total-profit function for the lawnmowing business is given by \(P(x)=9 x-250\) Find a function for the total revenue from mowing \(x\) lawns. How much does Jimmy charge per lawn? c) How many lawns must Jimmy mow before he begins making a profit?

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Find the domain of each function given below. $$g(x)=\sqrt{4+5 x}$$
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