Problem 596 Simplify. $$\left(u v^{-3}\rig... [FREE SOLUTION]

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Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 596

Simplify. $$\left(u v^{-3}\right)\left(u^{-4} v^{-2}\right)$$

Short Answer

Expert verified

Simplified expression: $ u^{-3} v^{-5} $

Step by step solution

Distribute the exponents

First, apply the exponents to each variable in the parentheses. This results in $ u v^{-3} $ and $ u^{-4} v^{-2} $.

Combine like bases

Next, combine the like bases: $ u $ terms together and $ v $ terms together. This gives us $ (u \times u^{-4}) (v^{-3} \times v^{-2}) $.

Use the exponents rule

Use the rule $ a^m a^n = a^{m+n} $ to combine the exponents. For the $ u $ terms: $ u^{1 - 4} = u^{-3} $. For the $ v $ terms: $ v^{-3 - 2} = v^{-5} $.

Combine the simplified expressions

Now, combine the simplified terms to get the final answer: $ u^{-3} v^{-5} $.

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Key Concepts

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

simplifying expressions

Simplifying expressions is an important part of prealgebra. It helps reduce complex expressions into simpler forms, making them easier to work with. To simplify an expression, you need to combine like terms and apply mathematical operations correctly. In our example, we simplified the expression \[\left(u v^{-3}\right)\left(u^{-4} v^{-2}\right)\] by breaking it down step-by-step. Each step is crucial for ensuring the final result is correct.

exponents

Exponents represent repeated multiplication of a number or variable by itself. In the expression \[\left(u v^{-3}\right)\left(u^{-4} v^{-2}\right)\], the exponents indicate how many times the base (either $u$ or $v$) is multiplied. For negative exponents, they indicate the reciprocal of the base raised to the corresponding positive exponent. For instance, $v^{-3}$ means $\frac{1}{v^3}$. Understanding how to work with exponents is key to simplifying expressions involving powers.

like bases

Like bases are terms in an expression that share the same base variable. In the example, we focus on combining the like bases $u$ and $v$.
When simplifying expressions with exponents, you group like bases together. Here, for $u$, we have $u \times u^{-4}$. For $v$, we have $v^{-3} \times v^{-2}$. This grouping allows us to apply exponent rules efficiently, ensuring our simplifications are accurate and straightforward.

exponent rules

Exponent rules are fundamental to simplifying expressions involving powers. One vital rule is $a^m \times a^n = a^{m+n}$, which states that you can add the exponents when multiplying like bases.
In the given example, we used this rule to combine the $u$ terms: $u^{1 - 4} = u^{-3}$, and the $v$ terms: $v^{-3 - 2} = v^{-5}$.
By applying this rule correctly, we simplified the original expression effectively to get the final answer: $u^{-3} v^{-5}$.

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91影视

Simplify. $$\left(u v^{-3}\right)\left(u^{-4} v^{-2}\right)$$

Short Answer

Step by step solution

Distribute the exponents

Combine like bases

Use the exponents rule

Combine the simplified expressions

Key Concepts

simplifying expressions

exponents

like bases

exponent rules

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