Problem 578 Simplify. $$\left(\frac{x}{x^{... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 578

Simplify. $$\left(\frac{x}{x^{9}}\right)^{5}$$

Short Answer

Expert verified

The simplified form is \ x^{-40} \.

Step by step solution

- Simplify the inner fraction

First, simplify the fraction inside the parentheses. The fraction is given as $\frac{x}{x^9}$. When dividing powers with the same base, subtract the exponents: $\frac{x}{x^9} = x^{1-9} = x^{-8}$.

- Apply the outer exponent

Now, raise the simplified expression to the power of 5. This means we calculate $ \big(x^{-8}\big)^5 $. When raising a power to another power, multiply the exponents: $(x^{-8})^5 = x^{-8 \times 5} = x^{-40}$.

- Final result

The simplified form of the given expression $ \left( \frac{x}{x^{9}} \right)^{5} $ is \ x^{-40} \.

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Key Concepts

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

simplifying expressions

When we simplify algebraic expressions, we aim to make them as simple as possible. Simplifying helps in solving equations and understanding the relationship between variables. It often involves combining like terms, factoring, or applying exponent rules. There are a few key steps:

Cancel out common factors in the numerator and denominator.
Apply algebraic properties such as the distributive property.
Utilize exponent rules like the power of a power or division of exponents.

In our exercise, we started by simplifying the fraction $\frac{x}{x^9}$. Since the base $x$ is the same in both the numerator and the denominator, we applied the division of exponents rule.

division of exponents

The division of exponents is a fundamental rule when working with algebraic expressions. When you divide powers that have the same base, you subtract the exponents. The rule is expressed as: \[ \frac{a^m}{a^n} = a^{m-n} \] where $a$ is the base and $m$ and $n$ are the exponents. In our example, we had $\frac{x}{x^9}$. Here, $x$ is the common base, and we subtract the exponent in the numerator from the exponent in the denominator: $\frac{x}{x^9} = x^{1-9} = x^{-8}$. This simplifies our fraction by reducing the exponents.

power of a power

The power of a power rule is used when you have an exponent raised to another exponent. The rule states that you multiply the exponents together: \[ (a^m)^n = a^{m \times n} \] This is especially useful for simplifying expressions with nested exponents. In our exercise, after simplifying the inner fraction to $x^{-8}$, we raised this expression to the power of 5: $(x^{-8})^5$. Using the power of a power rule, we multiply the exponents: $x^{-8 \times 5} = x^{-40}$. Thus, the final simplified form of the given expression is $x^{-40}$. Understanding this rule helps manage more complex expressions efficiently.

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91影视

Simplify. $$\left(\frac{x}{x^{9}}\right)^{5}$$

Short Answer

Step by step solution

- Simplify the inner fraction

- Apply the outer exponent

- Final result

Key Concepts

simplifying expressions

division of exponents

power of a power

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