Problem 29 - Algebra A Combined Function

Question

Write with positive exponents. Simplify if possible. $$ 8^{-4 / 3} $$

Step-by-Step Solution

Answer

The expression with positive exponents is \( \frac{1}{16} \).

Step 1: Rewrite as a Fractional Exponent

We start solving this by expressing the negative fractional exponent, so we have the expression \( 8^{-4/3} \). The negative exponent tells us to take the reciprocal of the base. Hence, \( 8^{-4/3} \) becomes \( \frac{1}{8^{4/3}} \).

Step 2: Simplify the Fractional Exponent

The next step involves simplifying the fractional exponent. The exponent \( 4/3 \) indicates the cube root and then raising to the fourth power. So, \( 8^{4/3} \) can be rewritten as \((8^{1/3})^4\). This says to take the cube root of 8 first, and then raise that result to the power of 4.

Step 3: Calculate the Cube Root of 8

Next, calculate the cube root of 8. Since 8 is \( 2^3 \), the cube root \( \sqrt[3]{8} \) yields 2. This means \( 8^{1/3} = 2 \).

Step 4: Raise the Result to the Fourth Power

Take the result from the previous step (which is 2) and raise it to the fourth power: \( 2^4 \). Therefore, we have \( 2^4 = 16 \).

Step 5: Write the Final Simplified Expression

Finally, write the expression in the form required by the problem statement: \( 8^{-4/3} = \frac{1}{16} \). Therefore, the expression with positive exponents is \( \frac{1}{16} \).

Key Concepts

Negative Exponents

Negative exponents might seem confusing initially, but they simply indicate the reciprocal of a number raised to the positive exponent. Consider an expression like \( a^{-n} \). This can be rewritten as \( \frac{1}{a^n} \). The negative sign in the exponent tells us to "flip" the base, turning it into its reciprocal.

This concept helps simplify expressions and makes it easier to deal with them. Remember:

• Transform \( x^{-m} \) into \( \frac{1}{x^m} \).
• Apply this rule for both whole and fractional exponents.
• Understanding negative exponents aids in simplifying complex algebraic expressions.

Negative exponents are also a gateway to comprehending other forms of exponents, such as fractional exponents.

Fractional Exponents

Fractional exponents are another way to represent roots. In the expression \( x^{m/n} \), \( n \) represents the root, and \( m \) indicates the power to which the root is raised. So, \( x^{m/n} \) is equivalent to \( (\sqrt[n]{x})^m \).

This dual relationship allows us to simplify and solve expressions effectively:

• \( x^{1/2} \) is the square root of \( x \), or \( \sqrt{x} \).
• \( x^{1/3} \) stands for the cube root of \( x \), or \( \sqrt[3]{x} \).
• When simplifying, always perform the root operation first, then raise the result to the power \( m \).

Using fractional exponents can dramatically simplify complex arithmetic and algebraic expressions, especially when dealing with powers and roots simultaneously.

Cube Root

The cube root of a number refers to the value that, when multiplied by itself twice, equals the original number. It is denoted by \( \sqrt[3]{x} \) or \( x^{1/3} \). This concept often appears in equations involving volumes and sciences because of its relationship with three-dimensional space.

Understanding cube roots involves recognizing:

• The cube root of \( 8 \) is \( 2 \), since \( 2 \times 2 \times 2 = 8 \).
• Cube roots can be fractional, as seen in \( \sqrt[3]{1} = 1 \).
• This operation is essential in simplifying expressions with higher order roots and powers.

Knowing how to calculate cube roots is crucial when you're dealing with expressions involving fractional exponents, as it simplifies the middle step of evaluating \( x^{m/n} \).

Simplification in Algebra

Simplification in algebra refers to the process of making algebraic expressions easier to work with. This involves performing operations such as factoring, expanding, and reducing expressions to their simplest form. Simplification helps ensure clarity and reduce the complexity of problems.

Key steps in algebraic simplification:

• Combine like terms where possible.
• Simplify expressions using exponent rules.
• Transform complex fractions and roots into simpler expressions.

In the context of exponents, turning negative and fractional exponents into simpler forms is a crucial stage of simplification, making it easier to handle expressions and solve equations. Practicing simplification regularly strengthens algebraic skills and aids in solving more advanced problems efficiently.