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Chapter 4: Problem 28

Simplify. Do not leave negative exponents in your answer. $$ y^{18} y^{-2} $$

Short Answer

Expert verified
y^{16}

Step by step solution

01

Apply the Product of Powers Property

Recall the Product of Powers Property: when multiplying two expressions with the same base, add the exponents. Here, both terms have the base y, so add the exponents 18 and -2.
02

Perform the Addition

Add the exponents: 18 + (-2) = 16. So, this simplifies to y^{16}.
03

Final Expression

Since 16 is a positive integer, the final simplified expression is y^{16}.

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Key Concepts

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

negative exponents
Negative exponents can be tricky, but with a bit of practice, they become easier to work with.

An exponent tells us how many times to multiply the base by itself. When the exponent is negative, we flip the base to its reciprocal (i.e., put the base in the denominator of a fraction) and then apply the positive exponent.

For instance, \( y^{-2} \) is the same as \( \frac{1}{y^2} \). This way, negative exponents are converted into positive exponents by changing their position in a fraction.
Note: Always try to simplify the expression so that you don鈥檛 leave any negative exponents in your answer.
product of powers property
The Product of Powers Property is a fundamental rule in exponents. This property states that when you multiply two expressions with the same base, you add their exponents.

Mathematically, it looks like this:
  • \( a^m \times a^n = a^{m+n} \)
For example, consider our exercise: \( y^{18} y^{-2} \). Both expressions have the same base 'y'. According to the Product of Powers Property, we add the exponents:
  • \( 18 + (-2) = 16 \)
Therefore, the expression \( y^{18} \times y^{-2} \) simplifies to \( y^{16} \).

Remember, this property only works if the bases of the expressions are the same.
adding exponents
Adding exponents is straightforward once you understand the rules. It is directly tied to the Product of Powers Property.

When dealing with expressions with the same base, all you have to do is add the exponents together. In our example, we have:
  • \(y^{18} \times y^{-2} \)
To add the exponents, you simply perform:
  • \(18 + (-2) = 16 \)
The simplified form becomes \( y^{16} \).

Always be careful with the signs of the exponents. If you have a negative exponent, remember to subtract it accordingly.
Sometimes, it helps to write out the addition clearly to avoid mistakes. Understanding these rules will make simplifying exponent expressions much easier!

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Most popular questions from this chapter

Waterfalls. In order for a waterfall to be classified as a classical waterfall, its height must be less than twice its crest width, and its crest width cannot exceed one-and-one-half times its height. The tallest waterfall in the world is about 3200 ft high. Let \(h\) represent a waterfall's height, in feet, and w the crest width, in feet. Write and graph a system of inequalities that represents all possible combinations of heights and crest widths of classical waterfalls.

Solve and graph. Write the answer using both set-builder notation and interval notation. $$ |x-4|+5>10 $$

Solve and graph. Write the answer using both set-builder notation and interval notation. $$ 25-2|a+3|>19 $$

Solve for x and y. Assume that a, b, c, d, and m are positive constants. $$ c(6 x-4)d $$

Solve and graph. Write the answer using both set-builder notation and interval notation. $$ |a+4| \leq 0 $$
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