Problem 256 In the following exercises, simp... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 256

In the following exercises, simplify. $$\frac{\left(y^{2}\right)^{5}}{y^{6}}$$

Short Answer

Expert verified

The simplified form is $ y^4 $.

Step by step solution

Apply the Power of a Power Rule

Use the rule $ (a^m)^n = a^{m \cdot n} $ to simplify $ (y^{2})^{5} $. This gives $ (y^{2})^{5} = y^{2 \cdot 5} = y^{10} $.

Apply the Quotient Rule

Use the quotient rule $ \frac{a^m}{a^n} = a^{m-n} $ to simplify $ \frac{y^{10}}{y^{6}} \.$ This gives $ \frac{y^{10}}{y^{6}} = y^{10-6} = y^{4} $.

Write the Final Answer

The simplified form of the given expression is $ y^4 $.

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Key Concepts

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

Power of a Power Rule

When simplifying expressions involving exponents, one useful rule is the power of a power rule. This rule helps when you raise an exponentiated number to another power. For example, in the expression $(y^2)^5$, you have an exponent raised to another exponent. According to the power of a power rule, we multiply the exponents. This rule is written as $ (a^m)^n = a^{m \cdot n} $. So, $(y^2)^5$ becomes $y^{2 \cdot 5} = y^{10}$.
The power of a power rule simplifies the calculation by reducing the nested exponents into a single exponent value. This makes the math easier and manageable.

Quotient Rule

Another crucial concept for simplifying expressions with exponents is the quotient rule. This rule comes in handy when you are dividing two numbers with the same base but different exponents, like in $\frac{y^{10}}{y^6}$.
The quotient rule states that $ \frac{a^m}{a^n} = a^{m-n} $, which means you subtract the exponent of the denominator from the exponent of the numerator. Following this rule, $\frac{y^{10}}{y^6}$ simplifies to $y^{10-6} = y^4$.
Using the quotient rule helps you simplify division problems involving exponents by reducing the number of operations.

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. In mathematical terms, $a^n$ means that the base $a$ is multiplied by itself \ times. For example, $2^3 = 2 \cdot 2 \cdot 2 = 8$.
There are several rules associated with exponents that make calculations easier:

Product Rule: $a^m \cdot a^n = a^{m+n}$
Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
Power of a Power Rule: $ (a^m)^n = a^{m \cdot n} $

Understanding and applying these rules correctly allow you to handle more complex expressions and simplify them efficiently.

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91影视

In the following exercises, simplify. $$\frac{\left(y^{2}\right)^{5}}{y^{6}}$$

Short Answer

Step by step solution

Apply the Power of a Power Rule

Apply the Quotient Rule

Write the Final Answer

Key Concepts

Power of a Power Rule

Quotient Rule

Understanding Exponents

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