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Chapter 4: Problem 86

\((10 y)^{9}(10 y)^{-8}\)

Short Answer

Expert verified
10y

Step by step solution

01

- Understand the Problem

You need to simplify the expression \( (10 y)^{9}(10 y)^{-8} \). This involves using the properties of exponents.
02

- Apply the Product of Powers Property

Use the property \(a^m \times a^n = a^{m+n}\) to combine the exponents of the same base. Here, the base is \(10y\).
03

- Add the Exponents

Add the exponents \(9\) and \(-8\): \( (10 y)^{9 + (-8)} = (10 y)^{1} \).
04

- Simplify the Expression

Simplify the final expression. \( (10 y)^{1} = 10 y\).

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Key Concepts

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

product of powers property
The Product of Powers Property is a fundamental rule in exponentiation. It states that when you multiply two expressions with the same base, you can simply add the exponents. Here's how it works: if you have a base, let's call it 'a,' raised to a power 'm' and the same base 'a' raised to a different power 'n,' multiplying these two can be simplified using the formula \(a^m \times a^n = a^{m+n}\).

This property makes working with exponents easier and is essential for simplifying expressions. In our exercise, the base is \(10y\). So, applying the property, we combine the exponents of \((10y)^{9}\) and \((10y)^{-8}\) by adding them together, resulting in \((10y)^{9+(-8)}\).
simplifying expressions
Simplifying expressions is crucial in mathematics to make equations or terms more manageable. In many cases, complicated expressions can be simplified using the properties of exponents. This involves:
  • Combining like terms.
  • Applying exponent rules such as the Product of Powers Property, Quotient of Powers Property, and Power of a Power Property.
  • Reducing fractions.


In our exercise, we start with the expression \((10 y)^{9}(10 y)^{-8}\). By using the Product of Powers Property, we combined the exponents into one single exponent. This makes the expression simpler, changing it to \((10 y)^{1}\), which further simplifies to just \(10 y\). Remember, simplifying expressions not only makes them easier to handle but also leads to more straightforward solutions.
exponent rules
Exponent rules are a set of guidelines that help you perform operations involving exponents more efficiently. Some essential rules include:
  • Product of Powers Property: \(a^m \times a^n = a^{m+n}\)
  • Quotient of Powers Property: \( \frac{a^m}{a^n} = a^{m-n}\) for \(a eq 0\)
  • Power of a Power Property: \((a^m)^n = a^{m \cdot n}\)


Using these rules makes solving mathematical problems involving exponents much easier. In our exercise, we specifically used the Product of Powers Property. We started with \((10 y)^{9}(10 y)^{-8}\) and applied the rule to combine the exponents. Adding the exponents, 9 and -8, we got 1, turning \((10 y)^{1}\) into \(10 y\). Knowing these rules can significantly simplify and speed up solving exponent-related problems.

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Most popular questions from this chapter

Perform each indicated operation. \(\left[\left(3 x^{2}-2 x+7\right)-\left(4 x^{2}+2 x-3\right)\right]-\left[\left(9 x^{2}+4 x-6\right)+\left(-4 x^{2}+4 x+4\right)\right]\)

Find each product. $$ (5 a+3 b)(5 a-4 b) $$

Find each product. $$ (2 m-3 n)(m+5 n) $$

\(\frac{\left(2 y^{-1} z^{2}\right)^{2}\left(3 y^{-2} z^{-3}\right)^{3}}{\left(y^{3} z^{2}\right)^{-1}}\)

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 103 \times 97 $$
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