Problem 81 For Exercises, simplify. $$\le... [FREE SOLUTION]

Chapter 9: Problem 81

For Exercises, simplify. $$\left(-5 a^{2}\right)\left(a^{-5}\right)^{2}$$

Short Answer

Expert verified

The simplified form of the given expression is $-\frac{5}{a^8}$

Step by step solution

Apply Power of a power rule

The power of a power rule states that to raise a power to a power, you multiply the exponents. Simplify $\left(a^{-5}\right)^{2}$ to $a^{-10}$ by multiplying the exponents.

Simplify the expression

Now we simplify the expression $-5a^2 * a^{-10}$. When we multiply exponential expressions having the same base, we add the exponents according to the rule $a^m * a^n = a^{m+n}$. In this case, $2 + (-10) = -8$. Thus, the simplified expression is $-5a^{-8}$.

Convert to positive exponent

We can write $-5a^{-8}$ as $-\frac{5}{a^8}$ as the negative exponent rule states that $a^{-n} = \frac{1}{a^n}$.

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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

The power of a power rule is a key concept in simplifying exponents. It states that when you raise a power to another power, you simply multiply the exponents. For example, consider the expression $(a^{-5})^2$. Instead of calculating the base multiple times, you multiply the exponents: $-5 \times 2 = -10$. Therefore, $(a^{-5})^2$ becomes $a^{-10}$.
This rule helps simplify complex expressions by reducing the number of steps required.

Negative Exponent Rule

The negative exponent rule is another important principle in algebra. It indicates how to transform expressions with negative exponents into more manageable forms. The rule simply states that $a^{-n} = \frac{1}{a^n}$.
Applying this in our example, $-5a^{-8}$ can be rewritten as $-\frac{5}{a^8}$. This transformation allows for easier algebraic manipulation and helps in understanding how exponential functions behave when they involve negative powers.

Exponential Expressions

Exponential expressions involve numbers raised to a power or exponent. Exponents represent how many times a number, known as the base, is multiplied by itself. For instance, in $a^2$, the base $a$ is multiplied by itself once.
Handling such expressions, especially when they involve multiple operations, requires an understanding of rules like the power of a power rule and the negative exponent rule. These rules allow us to transform and simplify expressions, helping us deal with complex formulas in everyday algebra.

Simplifying Algebraic Expressions

Simplifying algebraic expressions makes them easier to handle and understand. The goal is to reduce the expression to its simplest form while maintaining equivalency.
In our example, we start with $-5a^2 * a^{-10}$. By using the property that states when multiplying with the same base, add the exponents ($a^m * a^n = a^{m+n}$), we get $-5a^{-8}$. Next, applying the negative exponent rule finalizes the simplification to $-\frac{5}{a^8}$.

Simplification is essential for solving equations efficiently.
It helps recognize patterns and solve problems more quickly.

Understanding these processes is vital for mastering algebra.

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91影视

For Exercises, simplify. $$\left(-5 a^{2}\right)\left(a^{-5}\right)^{2}$$

Short Answer

Step by step solution

Apply Power of a power rule

Simplify the expression

Convert to positive exponent

Key Concepts

Power of a Power Rule

Negative Exponent Rule

Exponential Expressions

Simplifying Algebraic Expressions

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