Problem 74 - Algebra 2

Question

Simplify each expression. \(100^{-\frac{3}{2}} \cdot 0.01^{\frac{3}{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(10^{-6}\)

Step 1: Convert Into Power of 10

Begin by converting 100 into \(10^2\) and 0.01 into \(10^{-2}\). So, the expression becomes \((10^2)^{-\frac{3}{2}} \cdot (10^{-2})^{\frac{3}{2}}\)

Step 2: Apply Power of a Power Rule

The power of a power rule for exponents states that when raising a power to a power, the exponents are multiplied. Thus, we rewrite the expression as \(10^{2*(-\frac{3}{2})} \cdot 10^{-2*\frac{3}{2}}\) which simplifies to \(10^{-3}\cdot 10^{-3}\)

Step 3: Apply Product of Powers Rule

The product of powers rule states that when multiplying powers of the same base, add the exponents. Thus, \(10^{-3}\cdot 10^{-3}\) simplifies to \(10^{-3-3}\) which is \(10^{-6}\)

Key Concepts

Power of a Power Rule

The "power of a power" rule in exponents is a fundamental principle that simplifies expressions where an exponentiated number is raised to another power. This rule states that when you have a power raised to another power, you can simplify it by multiplying the exponents together.
For example, when you see \((a^m)^n\), the simplified form becomes \((a^{m \cdot n})\). This helps to quickly break down expressions that might seem complex at first.
Here's a step-by-step breakdown to grasp it:

• Identify the base and its exponent inside the parentheses.
• Observe the exponent outside the parentheses.
• Multiply the two exponents, as \((a^m)^n = a^{m\cdot n}\).

In the context of the original exercise, applying this rule allows us to simplify \((10^2)^{-\frac{3}{2}}\) to \((10^{2 \cdot -\frac{3}{2}} = 10^{-3})\).

Product of Powers Rule

Understanding the "product of powers" rule is essential when you are dealing with expressions that involve multiplying powers with the same base. This rule states that when multiplying like bases, you simply add their exponents together.
For instance, if you have an expression like \(a^m \cdot a^n\), you can rewrite it as \(a^{m+n}\). This makes long expressions manageable and easier to compute.
Steps to apply the product of powers rule include:

• Identify terms with the same base in the expression.
• Add their exponents together.
• Write down the base with the new exponent.

In the problem provided, using the product of powers rule translates \(10^{-3} \cdot 10^{-3}\) into \(10^{-3+(-3)} = 10^{-6}\).

Negative Exponents

Negative exponents can initially seem tricky, but they are straightforward once you get the hang of them. When you have a negative exponent, it indicates that the base is on the wrong side of a fraction line compared to a positive exponent.
The rule for negative exponents is: \(a^{-m} = \frac{1}{a^m}\). It transforms a negative exponent into a reciprocal of the positive exponent. This is useful for simplifying fractions and understanding equations better.
Consider these steps for dealing with negative exponents:

• Identify the negative exponent in the expression.
• Transform it into a fraction by taking the reciprocal.
• Ensure all other parts of the expression are consistent with this change.

In our example, the simplified expression \(10^{-6}\) is left as is, but knowing \(10^{-6} = \frac{1}{10^6}\) helps understand its magnitude and effect better.