Problem 21 - Algebra and Trigonometry

Question

Evaluate the expression using \(x=3, y=4,\) and \(z=-1\). \((9 x)^{2 / 3}+(2 y)^{2 / 3}+z^{2 / 3}\)

Step-by-Step Solution

Answer

The evaluated expression equals 14.

Step 1: Substitute the Values

Replace the variables with their given values. Substitute \(x = 3\), \(y = 4\), and \(z = -1\) into the expression: \[(9x)^{2/3} + (2y)^{2/3} + z^{2/3}\]This becomes:\[(9 \times 3)^{2/3} + (2 \times 4)^{2/3} + (-1)^{2/3}\]

Step 2: Simplify Each Term

Calculate each part of the expression:1. \((9 \times 3) = 27\), then \(27^{2/3}\).2. \((2 \times 4) = 8\), then \(8^{2/3}\).3. \((-1)^{2/3}\).

Step 3: Calculate the Powers

Evaluate the fractional exponents:1. \(27^{2/3}\): - First, find the cube root: \(\sqrt[3]{27} = 3\). - Then, square the result: \(3^2 = 9\).2. \(8^{2/3}\): - First, find the cube root: \(\sqrt[3]{8} = 2\). - Then, square the result: \(2^2 = 4\).3. \((-1)^{2/3}\): - First, find the cube root: \(\sqrt[3]{-1} = -1\). - Then, square the result: \((-1)^2 = 1\).

Step 4: Sum the Results

Add all simplified components of the expression:- From Step 3, the terms are \(9\), \(4\), and \(1\).- Add them together: \(9 + 4 + 1 = 14\).

Key Concepts

Fractional Exponents

Fractional exponents are commonly seen in algebra, and they can be a bit tricky at first. But don't worry! Understanding them is actually pretty simple once you break it down. A fractional exponent like \( a^{m/n} \) signifies two operations:

* Taking the \( n^{th} \) root (denominator). * Raising to the \( m \) power (numerator).

For example, for \( 27^{2/3} \):

• Start by Finding the Root:

  First, determine the cube root (since the denominator is 3). The cube root of 27 is 3 because \( 3 \times 3 \times 3 = 27 \).

• Then Square the Result:

  Next, square the 3 from the cube root, which gives you 9 as \( 3^2 = 9 \).

Always remember, it's like pulling apart your favorite sandwich and enjoying each layer separately, before putting it all together to appreciate the full taste!

Substituting Values

Substituting values is like replacing placeholders with actual numbers in a recipe. Imagine if you're cooking with precise measurements! When working with algebraic expressions, we input specific numbers given for each variable to make calculations clearer.

Given the values \(x = 3, y = 4,\) and \(z = -1\), we substitute these into the expression \((9x)^{2/3} + (2y)^{2/3} + z^{2/3}\). Here's what it looks like:

• Replace the Variables:

  Instead of "\(x\)", use 3. For "\(y\)", use 4. Replace "\(z\)" with -1.

• Expression Becomes:

  Calculating each component gives us:
  • \((9 \times 3)^{2/3}\)
  • \((2 \times 4)^{2/3}\)
  • \((-1)^{2/3}\)

Substituting values is just about putting the puzzle pieces in the right places to get the full picture!

Simplifying Algebraic Expressions

Simplifying algebraic expressions might sound complicated, but it’s really about making things clearer and easier to understand. Our goal is to break down complex expressions into simpler parts and find a single value if needed.

The expression \((9 \times 3)^{2/3} + (2 \times 4)^{2/3} + (-1)^{2/3}\) provides a great example. To simplify:

• Calculate Inside the Parentheses:

  • First, solve \(9 \times 3\) which gives 27.
  • Then, solve \(2 \times 4\) which gives 8.

• Simplify Using Fractional Exponents:

  • \(27^{2/3} \rightarrow 9\) (as explained in fractional exponents section)
  • \(8^{2/3} \rightarrow 4\)
  • \((-1)^{2/3} \rightarrow 1\)

• Add all Simplified Terms Together:

  This results in: 9 + 4 + 1 = 14.

The key to simplifying is not to rush. Look at each part one at a time, and slowly watch everything fall into place.