Problem 117 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \left(-27 x^{6}\right)^{-1 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{-3x^2}\).

Step 1: Apply Negative Exponent Rule

The negative exponent rule states that \(a^{-n} = \frac{1}{a^n}\). Here, we have an expression \((-27x^6)^{-1/3}\). According to the rule, this becomes \(\frac{1}{(-27x^6)^{1/3}}\).

Step 2: Simplify Using the Cube Root

Now, we need to simplify \((-27x^6)^{1/3}\). The cube root of a product is the product of the cube roots: \((-27)^{1/3} \times (x^6)^{1/3}\).

Step 3: Simplify Cube Root of Term with a Coefficient

Find the cube root of \(-27\). Since \(-3\times -3\times -3 = -27\), we have \((-27)^{1/3} = -3\).

Step 4: Simplify Cube Root of Power of a Variable

Simplify \((x^6)^{1/3}\). The property \((a^m)^{n} = a^{mn}\) applies, and \((x^6)^{1/3} = x^{6/3} = x^2\).

Step 5: Combine Results

Combine the results from Steps 3 and 4: \( (-27)^{1/3} \times (x^6)^{1/3} = -3 \times x^2 = -3x^2\).

Step 6: Simplify the Main Expression

Substitute the result from Step 5 back into the fraction from Step 1: \( \frac{1}{-3x^2}\). This is the simplified form of the expression.

Key Concepts

Negative Exponent Rule

In algebra, you might encounter exponents that are negative. This situation often prompts a bit of head-scratching for students, but don’t worry. The negative exponent rule is your go-to tool for simplifying these expressions. The rule states that any base with a negative exponent can be rewritten as a fraction:

• If you have an expression like \[a^{-n}\], you can transform it to \[\frac{1}{a^n}\].

This rule applies regardless of whether \(a\) is a number or a variable. By applying this rule to an expression, you convert an intimidating negative exponent into a positive one, making further simplification straightforward.
For example, if you have \((-27x^6)^{-1/3}\), applying the negative exponent rule changes it to \[\frac{1}{(-27x^6)^{1/3}}\]. This transformation sets the stage for additional simplifications, such as cube roots.

Cube Root Simplification

Taking the cube root of a number or expression means finding a value that, when multiplied by itself three times, gives you the original number or expression. Simplifying cube roots requires breaking down complex terms into simpler ones. Here’s how you can tackle them:

• For coefficients (the numbers attached to variables), find a number that multiplied by itself three times equals the coefficient.
  • For \((-27)\), the cube root is \(-3\), because \(-3 \times -3 \times -3 = -27\).
• For variables with exponents, use the property \[(a^m)^n = a^{m \times n}\] to simplify.
  • For example, \((x^6)^{1/3}\) simplifies to \(x^{6/3}\).
  • Breaking it down further, \(x^{6/3} = x^2\).

Understanding these simplifications allows quicker and easier handling of complex algebraic expressions, and ensures you can solve similar problems independently.

Properties of Exponents

Exponents are not just numbers sitting above base numbers; they have properties that allow mathematical expressions to be manipulated and simplified efficiently. Knowing the key properties of exponents enhances your ability to handle algebraic expressions:

• Product of Powers Property: \[a^m \times a^n = a^{m+n}\]. This property lets you combine powers with the same base.
• Power of a Power Property: \[(a^m)^n = a^{m \times n}\]. Useful for dealing with expressions like \((x^6)^{1/3}\).
• Quotient of Powers Property: \[\frac{a^m}{a^n} = a^{m-n}\]. Simplify expressions where you divide like bases.
• Zero Exponent Rule: Any base raised to the power of zero is \(1\) (e.g., \[a^0 = 1\]).

Using these properties allows mathematicians and students alike to simplify and solve expressions more efficiently. In our exercise, we specifically used the power of a power property to simplify the cube root of a variable raised to a power. Keeping these rules in mind helps you decipher any algebraic challenge with greater ease.