Problem 555 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(1000^{\frac{2}{3}}\) (b) \(25^{\frac{3}{2}}\) (c) \(32^{\frac{3}{5}}\)

Step-by-Step Solution

Answer

(a) 100, (b) 125, (c) 8

Step 1 - Simplifying the first expression

Consider the expression: \(1000^{\frac{2}{3}}\). First, rewrite 1000 as a power of 10: \(1000 = 10^3\). Then the expression becomes: \[1000^{\frac{2}{3}} = (10^3)^{\frac{2}{3}} = 10^{3 \cdot \frac{2}{3}} = 10^2 = 100\] Therefore, \(1000^{\frac{2}{3}} = 100\).

Step 2 - Simplifying the second expression

Consider the expression: \(25^{\frac{3}{2}}\). First, rewrite 25 as a power of 5: \(25 = 5^2\). Then the expression becomes: \[25^{\frac{3}{2}} = (5^2)^{\frac{3}{2}} = 5^{2 \cdot \frac{3}{2}} = 5^3 = 125\] Therefore, \(25^{\frac{3}{2}} = 125\).

Step 3 - Simplifying the third expression

Consider the expression: \(32^{\frac{3}{5}}\). First, rewrite 32 as a power of 2: \(32 = 2^5\). Then the expression becomes: \[32^{\frac{3}{5}} = (2^5)^{\frac{3}{5}} = 2^{5 \cdot \frac{3}{5}} = 2^3 = 8\] Therefore, \(32^{\frac{3}{5}} = 8\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent. When a number is raised to a power, it means multiplying the base by itself as many times as indicated by the exponent. For example, in the expression \(10^3\), 10 is the base, and 3 is the exponent, which means:

• 10 x 10 x 10 = 1,000

This concept helps simplify larger calculations, as seen with multiplying the same number multiple times. It's essential to understand basic exponentiation to work efficiently with more complex topics, like fractional exponents and powers of numbers.

Fractional Exponents

Fractional exponents extend the idea of exponentiation. They can represent both rooting and regular powers. For example, the expression \(1000^{\frac{2}{3}}\) involves finding both a root and raising to a power. Here's a breakdown:

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• First, we rewrite 1000 as a power of 10: \(1000 = 10^3\).
  
• Then, we apply the fractional exponent: \(1000^{\frac{2}{3}} = (10^3)^{\frac{2}{3}}\).
  
• Multiplying exponents, we get: \(10^{3 \cdot \frac{2}{3}} = 10^2\) which equals 100.
  

Likewise, for the other expressions:

• \(25^{\frac{3}{2}}\) simplifies by understanding \(25 = 5^2\).
  
• Applying the exponent: \(25^{\frac{3}{2}} = (5^2)^{\frac{3}{2}} = 5^{3}\) equals 125.
  
• For \(32^{\frac{3}{5}}\), knowing \(32 = 2^5\) helps simplify to \(2^3 = 8\).
  

Understanding fractional exponents makes simplifying and calculating these expressions manageable.

Powers of Numbers

Powers of numbers are a way of expressing repeated multiplication. In simpler terms, raising a number to a specific power multiplies the number by itself a certain number of times. This is crucial because certain numbers take specific forms that are easier to work with, such as:

• 1000 as \(10^3\)
  
• 25 as \(5^2\)
  
• 32 as \(2^5\)
  

Recognizing these forms allows us to simplify exponential expressions quickly. Using the steps:

• Convert the base into its power equivalent.
  
• Apply the exponent rules.
  
• Multiply the exponents and simplify.
  

This approach aids in handling more complex calculations by breaking them into more digestible parts.