Problem 27 - Algebra and Trigonometry

Question

\(25-28\) . 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 is 14.

Step 1: Substitute the Values of Variables

Replace each variable in the expression with its given value. The expression becomes:\[(9 \times 3)^{2/3} + (2 \times 4)^{2/3} + (-1)^{2/3}\]

Step 2: Simplify Inside the Parentheses

Calculate the values inside the parentheses:1. For \(9 \times 3\), calculate \(9 \times 3 = 27\).2. For \(2 \times 4\), calculate \(2 \times 4 = 8\).3. The last term is already \(-1\). Now the expression is:\[27^{2/3} + 8^{2/3} + (-1)^{2/3}\]

Step 3: Evaluate the Exponents

Now calculate each term with the fractional exponent:1. \(27^{2/3}\): The cube root of 27 is 3, so \((27^{1/3} = 3)\), and \(3^2 = 9\).2. \(8^{2/3}\): The cube root of 8 is 2, so \((8^{1/3} = 2)\), and \(2^2 = 4\).3. \((-1)^{2/3}\): The cube root of -1 is -1, and \((-1)^2 = 1\).

Step 4: Add the Simplified Terms

Sum the evaluated terms from the previous step:\(9 + 4 + 1 = 14\).

Key Concepts

Substitution in Expressions

When you encounter an algebraic expression, substitution is a key step in evaluating it. Substitution involves replacing each variable in the expression with a given numeric value.
For example, if you're given an expression like \((9x)^{2/3} + (2y)^{2/3} + z^{2/3}\), and you know that \(x=3\), \(y=4\), and \(z=-1\), you'll substitute these values wherever the variables appear.
By doing this, you turn the expression into numerical form, like so: \((9 \times 3)^{2/3} + (2 \times 4)^{2/3} + (-1)^{2/3}\).

• Step-by-step substitution:
  • Replace \(x\) with 3, \(y\) with 4, and \(z\) with -1.
  • This results in the values in parentheses changing to actual numbers.

Once all the variables are replaced with their respective values, you can proceed to simplify the expression further.

Simplifying Exponents

After substituting the values, you will often need to simplify expressions that contain exponents. Simplifying exponents means performing any arithmetic inside the base of the exponents and then addressing the exponent.
For example, consider the expression \( (9 \times 3)^{2/3} + (2 \times 4)^{2/3} + (-1)^{2/3} \). First, calculate what is inside the parentheses:

• Calculate \(9 \times 3\), which equals 27.
• Calculate \(2 \times 4\), which equals 8.
• \(-1\) is already simplified.

Once you have these simplified bases (27, 8, -1), you can then apply the exponents.

Fractional Exponents

Fractional exponents represent both a root and a power simultaneously. To evaluate them, you typically take the root first and then apply the power.

• Understanding Fractional Exponents: In \(a^{m/n}\), \(n\) signifies the root, and \(m\) signifies the power.
• In this problem, look at expressions like \(27^{2/3}\):
• First, take the cube root of 27, which is 3.
• Then, raise that result to the power of 2, giving you 9.

Similarly, you can calculate \(8^{2/3}\):

• The cube root of 8 is 2.
• Square that result to get 4.

Negative numbers like \((-1)^{2/3}\) require careful handling:

• The cube root of -1 is -1.
• But when squared, it becomes positive, resulting in 1.

Once you've simplified each fractional exponent, you can add the terms together for your final result.