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Chapter 10: Problem 62

Simplify. $$16^{-1 / 2}$$

Short Answer

Expert verified
\(16^{-1/2} = \frac{1}{4}\)

Step by step solution

01

Understand the Problem

The expression to simplify is \(16^{-1/2}\). This involves understanding how to deal with negative exponents and roots in the expression.
02

Rewrite the Negative Exponent

Rewrite the expression \(16^{-1/2}\) as a reciprocal to handle the negative exponent: \(\frac{1}{16^{1/2}}\).
03

Simplify the Radical

Simplify \(16^{1/2}\) by recognizing that taking the square root of 16 is the same as raising it to the power of \(1/2\). So \(16^{1/2} = \sqrt{16} = 4\).
04

Compose the Final Answer

Now that \(16^{1/2} = 4\), the expression becomes \(\frac{1}{4}\).

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Key Concepts

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

negative exponent
Negative exponents might seem tricky at first, but they're easy once you understand the key idea.
Instead of multiplying a number, a negative exponent tells you to take the reciprocal of that number and then apply the positive power.
For example, instead of raising to the power of (-1/2), use the positive power (1/2) and take the reciprocal.
So, (16^{-1/2} becomes (1 / 16^{1/2})
This gives us a simpler way to handle and understand expressions with negative exponents.
reciprocal
A reciprocal of a number is simply 1 divided by that number.
For example, the reciprocal of 4 is (1/4), and the reciprocal of (a/b) is (b/a).
Handling reciprocals is especially useful when dealing with negative exponents, because a negative exponent indicates we need the reciprocal.
So when you see (16^{-1/2}, rewriting it as a reciprocal transforms it into (1/16^{1/2}).
Once you have it as a reciprocal, you can proceed to simplify further.
square root
The square root is a special exponent - it鈥檚 the same as raising a number to the power of 1/2.
For example, the square root of 16 is (4) because (4 * 4 = 16).
When we simplify (16^{1/2}, we are essentially asking: What number multiplied by itself gives us 16? The answer is 4.
So, (16^{1/2} = 4).
With this understanding, you can easily simplify expressions with square roots, making calculation straightforward and more manageable.

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Most popular questions from this chapter

Factor completely. $$ 9 x^{2} y^{2}-30 x y+25 $$

How can you tell from the equation of an ellipse whether its graph is horizontal or vertical?

The standard form for equations of horizontal or vertical hyperbolas centered at \((h, k)\) are as follows: $$ \frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1 $$ (Graph can't copy) $$ \frac{(y-k)^{2}}{b^{2}}-\frac{(x-h)^{2}}{a^{2}}=1 $$ The vertices are as labeled and the asymptotes are $$ y-k=\frac{b}{a}(x-h) \text { and } y-k=-\frac{b}{a}(x-h) $$ For each of the following equations of hyperbolas, complete the square, if necessary, and write in standard form. Find the center, the vertices, and the asymptotes. Then graph the hyperbola. $$ \frac{(x-5)^{2}}{36}-\frac{(y-2)^{2}}{25}=1 $$

Classify each of the following as the equation of either a circle, an ellipse, a parabola, or a hyperbola. $$ 16 x^{2}+5 y^{2}-12 x^{2}+8 y^{2}-3 x+4 y=568 $$

Solve. $$x^{2}+4 x=60$$
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