Problem 36 - Elementary and Intermediate Algebra

Question

Simplify each expression. Assume that the variables can be any real number, and use absolute value symbols when necessary. See Example 2 . $$ \left(-64 t^{9}\right)^{1 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is \(-4t^3\).

Step 1 - Apply the Cube Root

The given expression is \((-64 t^9)^{1/3}\). To simplify the expression, begin by taking the cube root, which means raising both the number \(-64\) and the variable part \(t^9\) to the power of \(1/3\).

Step 2 - Simplify the Cube Root of the Coefficient

The cube root of \(-64\) is \(-4\) because \((-4)^3 = -64\). Therefore, \((-64)^{1/3} = -4\).

Step 3 - Simplify the Cube Root of the Variable

To simplify \((t^9)^{1/3}\), use the property \((a^m)^{n} = a^{m \, n}\). Therefore, \(t^{9 \, (1/3)} = t^{9/3} = t^3\).

Step 4 - Combine the Results

Substitute the simplified parts back into the expression: \((-4)\cdot(t^3) = -4t^3\).

Key Concepts

Understanding Cube Roots

The cube root is an interesting mathematical operation. It helps us figure out which number, when multiplied by itself three times, gives us the original number. Just like how the square root is for squares, the cube root is for cubes. It's represented by the symbol \( \sqrt[3]{x} \). Let's break this down:

• For positive numbers, like 8, the cube root is simply 2 because \( 2 \times 2 \times 2 = 8 \).
• For negative numbers, the cube root is also negative. Take \(-64\) for example, where the cube root is \(-4\) because \(-4 \times -4 \times -4 = -64\).

The cube root is unique because it works smoothly with negative numbers, unlike the square root, which stays within positive realms or imagines a new dimension for negatives.
Taking the cube root of any variable expression, like \( t^9 \), involves applying the rule \( (a^m)^n = a^{m \times n} \). This helps us simplify expressions like \( (t^9)^{1/3} \) to \( t^3 \). So, simply put, the cube root can be thought of as the opposite of cubing a number or variable. It's about finding the original factor that would give rise to a cubic expression.

Exploring Exponents

Exponents are a powerful tool in mathematics that allows us to concise how many times a number or variable is multiplied by itself. When you see something like \( t^9 \), it means we multiply \( t \) by itself 9 times.

• Exponents make expressions shorter and easier to handle. Instead of writing 9 \( t\)'s out in a row, we just use \( t^9 \).
• When we have \( t^{9/3} \), we apply the rule \( a^{m/n} = \sqrt[n]{a^m} \). This simplifies calculations and expressions greatly.
• Another handy rule is \( (a^m)^n = a^{m \times n} \), letting us compute higher powers in a straightforward manner.

Exponents interact in versatile ways with mathematical operations, making them essential to algebraic simplification. They're like the language of powers, conversing through roots and simplified expressions.

Understanding Absolute Value

The concept of absolute value is simple yet incredibly important in mathematics. It answers the question: how far is a number from zero, without caring about its direction?

• Absolute value is represented by vertical bars, like \( |x| \), and is always non-negative.
• For any real number \( x \), if \( x \geq 0 \), then \( |x| = x \). If \( x < 0 \), then \( |x| = -x \).
• This means \( |-3| = 3 \) and \( |3| = 3 \). Both numbers sit three units away from zero on the number line.

Absolute value helps ensure that distance and magnitude are positive, which is useful when variables can take on any real number in an expression. It's crucial in problems where direction doesn't matter, only magnitude. Often, when simplifying expressions involving variables, we might need to use absolute values to express the result correctly—especially in cases involving roots or fractional exponents.