Problem 42 - College Algebra

Question

Simplify each exponential expression $$ \left(-\frac{6}{y}\right)^{3} $$

Step-by-Step Solution

Answer

-216 / \( y^3 \)

Step 1: Analyze the problem

The base is -6/y and the exponent is 3. The laws of exponents state that an exponent signifies how often to use the number in a multiplication. Here, this means we have to multiply the base -6/y three times.

Step 2: Multiply the base three times

(-6/y) * (-6/y) * (-6/y). Multiplication of fractions is carried out by multiplying the numerators together to get the numerator of the product and multiplying the denominators together to get the denominator of the product.

Step 3: Simplify

On multiplication, we will end up with (-6*-6*-6) / (y*y*y) = -216 / y^3. In the numerator, -6 cubed gives -216 and in the denominator, y cubed remains as y^3.

Key Concepts

Laws of Exponents

Understanding the laws of exponents is crucial when dealing with exponential expressions. Exponents are shorthand for repeated multiplication. If you have a number like \((-\frac{6}{y})^3\), the exponent tells you how many times to multiply the base by itself. Here, the base is \(-\frac{6}{y}\) and the exponent is 3, which means we multiply \(-\frac{6}{y}\) by itself three times. Using the laws of exponents helps simplify expressions, especially when dealing with negative bases or fractions. These laws include the product of powers, power of a power, and power of a product rules. When these laws are applied correctly, they simplify calculations and make complex problems manageable.

Multiplication of Fractions

Multiplying fractions might seem tricky at first, but it's a straightforward process. To multiply fractions, follow these simple steps:

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.

For example, in the expression \((-\frac{6}{y})\), multiplying it three times involves multiplying the top parts \((-6)\) by each other, and the bottom part \(y\) three times. So, \((-6) \times (-6) \times (-6)\) will give the new numerator, and \(y \times y \times y\) will be the denominator. This results in a simplified fraction. Be mindful of negative signs. In this case, multiplying an odd number of negatives results in a negative product. So, the resulting expression is \(-\frac{216}{y^3}\).

Exponentiation

Exponentiation is the mathematical operation involving two numbers, the base, and the exponent. It indicates how many times to multiply the base by itself. This is what occurs when we simplify \((-\frac{6}{y})^3\). The process of exponentiation for a fractional base involves treating the numerator and denominator separately.

• Raise each part within the fraction to the given power separately.
• Combine results to form the exponentiated expression.

For example, raising \(-6\) to the third power gives \(-6 \times -6 \times -6 = -216\), and raising \(y\) to the third power gives \(y \times y \times y = y^3\). Thus, exponential expressions with fractions require careful attention to both the signs and the operations on numerators and denominators.

Simplification of Expressions

Simplifying expressions means reducing them to their most basic form. It often makes them easier to work with or understand. For this exercise, after performing the exponentiation and fraction multiplication, we simplify the expression \((-\frac{6}{y})^3\) to \(-\frac{216}{y^3}\). Simplification involves combining like terms, reducing fractions, and relying on basic arithmetic principles.

• Check if the numerical part can be reduced or combined.
• Ensure that common terms in the numerator and denominator cancel out whenever possible.

In this case, the multiplication does not have further common terms, but simplifying the expression helps in recognizing the structure: a negative numerator and an exponentiated denominator. Ensuring every step is correct allows for an accurate final expression. Simplification is crucial in making expressions more manageable in algebraic manipulations.