Problem 54 - College Algebra with Modeling and Visualization

Question

Exercises \(55-64:\) Use the power rules to simplify the expression. Use positive exponents to write your answer. $$ \left(5^{-1}\right)^{3} $$

Step-by-Step Solution

Answer

\( \left(5^{-1}\right)^{3} = \frac{1}{125} \)

Step 1: Understand the Power of a Power Rule

The power of a power rule states that when you have an expression like \((a^m)^n\), it can be simplified by multiplying the exponents: \((a^m)^n = a^{mn}\). In this exercise, you have \((5^{-1})^3\).

Step 2: Apply the Power of a Power Rule

Apply the rule by multiplying the exponents: \((-1) \times 3 = -3\). Therefore, \((5^{-1})^3 = 5^{-3}\).

Step 3: Convert Negative Exponent to Positive Exponent

Recall that a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. Hence, \(5^{-3} = \frac{1}{5^3}\).

Step 4: Calculate the Positive Exponent

Now, calculate \(5^3\). This is calculated as \(5 \times 5 \times 5 = 125\). Therefore, \(5^{-3} = \frac{1}{125}\).

Key Concepts

Exponentiation

Exponentiation is a fundamental concept in algebra, involving raising a base number to the power of an exponent. It's important to understand that the base is the number being multiplied, and the exponent indicates how many times it is multiplied by itself. So, when you see an expression like \(a^b\), it means \(a\) multiplied by itself \(b\) times.
To simplify complicated expressions involving multiple exponential operations, algebra employs several rules. These include the product of powers rule, power of a power rule, and power of a product rule. Each of these rules assists in managing different forms of expressions to make calculations easier.
Another key aspect of exponentiation is understanding that raising a number to the power of 1 returns the number itself, while any number raised to the power of 0 equals 1."},{"concept_headline":"Negative Exponents","text":"Negative exponents can initially be confusing, but they are just as straightforward as positive exponents once you grasp the concept. A negative exponent suggests taking the reciprocal of the base and then applying the positive exponent.
For example, if you come across an expression like \(5^{-3}\), you're dealing with the reciprocal of \(5^3\). Essentially, negative exponents flip the fraction, turning it inside out. So, \(5^{-3} = \frac{1}{5^3}\).
Understanding this reciprocal transformation is crucial for simplifying expressions with negative exponents, especially down to their positive counterparts."},{"concept_headline":"Simplifying Expressions","text":"Simplifying expressions in algebra can often feel like solving a complex puzzle, but it becomes easier with a few straightforward steps and an understanding of basic rules.

• Identify and apply the algebraic rules applicable to the expression. For example, if you have an exponent of another exponent, apply the power of a power rule.
• Use the rules of exponents consistently, including converting negative exponents into positive ones by taking the reciprocal.
• Perform any necessary calculations to reduce the expression to its simplest form. This often includes multiplying or dividing the terms under the new and simpler exponents.

By breaking down the expression systematically and using these rules, you can simplify complex expressions into a more manageable form. In the case of our earlier example, the original expression \((5^{-1})^3\), uses these principles to eventually simplify it as \(\frac{1}{125}\)."}]}]}]}]}}]}]}]}}]}]}]}]}]}]}]}]}]}}]}]}]}]}]}}]}]}]}]}]}]}]}]}]}]}]}

Negative Exponents

Negative exponents can initially be confusing, but they are just as straightforward as positive exponents once you grasp the concept. A negative exponent suggests taking the reciprocal of the base and then applying the positive exponent.
For example, if you come across an expression like \(5^{-3}\), you're dealing with the reciprocal of \(5^3\). Essentially, negative exponents flip the fraction, turning it inside out. So, \(5^{-3} = \frac{1}{5^3}\).
Understanding this reciprocal transformation is crucial for simplifying expressions with negative exponents, especially down to their positive counterparts.

Simplifying Expressions

Simplifying expressions in algebra can often feel like solving a complex puzzle, but it becomes easier with a few straightforward steps and an understanding of basic rules.

• Identify and apply the algebraic rules applicable to the expression. For example, if you have an exponent of another exponent, apply the power of a power rule.
• Use the rules of exponents consistently, including converting negative exponents into positive ones by taking the reciprocal.
• Perform any necessary calculations to reduce the expression to its simplest form. This often includes multiplying or dividing the terms under the new and simpler exponents.

By breaking down the expression systematically and using these rules, you can simplify complex expressions into a more manageable form. In the case of our earlier example, the original expression \((5^{-1})^3\), uses these principles to eventually simplify it as \(\frac{1}{125}\).