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Exponents Study Guide (page 2)

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Updated on Oct 3, 2011

Negative Exponents

So far, all of the exponents we have seen in this lesson have been positive. But we can work with negative exponents, too. Remember: a positive number multiplied by a negative number will give you a negative product, so a positive exponent multiplied by a negative exponent will give you a negative exponent:(x3)–5 = x–15, because (3)(–5) = –15.

Example

What is (d–3)–9?
A negative number multiplied by a negative number will give you a positive product: (–3)( –9) = 27, so (d–3)–9 = d27.

Previously, we learned that bases with negative exponents appear as positive exponents in the denominator when the term is expressed as a fraction. When a coefficient is raised to a negative exponent, express the result as a fraction. Make the numerator of the fraction 1, and place the base in the denominator with a positive exponent:4–2 is equal to , which is .

Example

What is (5m3)–4?
5m3 is in parentheses, so the entire term is raised to the negative fourth power: (3)(–4) = –12, so (m3)–4 = m–12. (5)–4 = , which is .
(5m3)–4 = , which could also be written as .

Zero Exponents

An expression with a term that has an exponent of 0 is the easiest to simplify. Any value—number or variable—raised to the power of 0 is 1: 10 = 1, 20 = 2, 1000 = 1, x0 = 1, and (abcd)0 = 1. We must still remember our order of operations though. The term 3x0 has only x raised to the power of zero, so this term is equal to 3(x0) = 3(1) = 3. Only the base that is raised to the zero power is equal to 1. If a term has a coefficient or another base that is not raised to the zero power, those parts of the term must still be evaluated.

Example

What is 4a0b2?
a0 = 1, so 4(a0)(b2) = 4(1)(b2) = 4b2

Fractional Exponents

A constant or variable can also be raised to a fractional exponent. The numerator of the exponent is the power to which the constant or variable is raised. The denominator of the exponent is the root that must be taken of the constant or variable. For instance x, means that x is raised to the third power, and then the second, or square, root of x must be taken. The root symbol, or radical symbol, looks like this: √. The square root of x is √x. To show that x is raised to the third power, we put the exponent inside the radical: √x3, therefore, = √x3.

If we needed to find a larger root of x, such as 5 or 6, that number would be placed just outside the radical symbol: x is the fifth root of x2 raised to the second power. We place x2 inside the radical and 5 outside the radical: x = 5√ x2.

Find practice problems and solutions for these concepts at Exponents Practice Questions.

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