Problem 35 - College Algebra

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. $$ \left(y^{7}\right)^{3} \div x^{14} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{y^{21}}{x^{14}} \).

Step 1: Apply Power of a Power Rule

The expression given is \( (y^7)^3 \). According to the power of a power rule, \( (a^m)^n = a^{m \times n} \), we multiply the exponents. So \( (y^7)^3 = y^{7 \times 3} = y^{21} \).

Step 2: Simplify Division of Same Base Exponents

Now you need to simplify \( y^{21} \div x^{14} \). Since there are no same-base exponents to divide, the expression remains as \( \frac{y^{21}}{x^{14}} \). This is already simplified and contains positive exponents.

Key Concepts

Power of a Power Rule

When dealing with exponents, the Power of a Power Rule is a fundamental concept that can simplify complex expressions quickly. This rule applies to terms that are raised by another exponent, like \((a^m)^n\). In such cases, you multiply the two exponents together, leading to a new power: \(a^{m \times n}\). Let's break it down with an example. Consider \((y^7)^3\). Applying the Power of a Power Rule means you multiply the exponents: 7 and 3. Consequently, \((y^7)^3\) becomes \(y^{7 \times 3} = y^{21}\). This rule helps simplify expressions and is essential for working efficiently with polynomial problems and expressions involving powers. It is a handy tool for making expressions more manageable and avoiding unnecessary calculations.

Exponents

Exponents are shorthand notation used in algebra to express repeated multiplication. When a number or variable is raised to a power, it indicates how many times to multiply the base by itself. For example, in \(y^7\), 7 is the exponent, which means \(y\) is multiplied by itself 7 times: \(y \times y \times y \times y \times y \times y \times y\). Important things to remember about exponents:

• When the exponent is 1, the base remains unchanged; \(a^1 = a\).
• Any number raised to the power of zero equals 1; \(a^0 = 1\), provided that \(a eq 0\).
• Positive exponents denote repeated multiplication, whereas negative exponents indicate division or reciprocal.

Understanding exponents is crucial for mastering higher-level math topics, including algebra and calculus, as they form the basis for operations involving growth and decay, areas, and volumes.

Division of Exponents

The Division of Exponents is one portion of exponential rules we often encounter in algebra. This rule states that when dividing two powers that have the same base, you subtract their exponents. The formula is \(a^m \div a^n = a^{m-n}\). For this rule to apply, the bases must be the same.However, in our example, \(y^{21} \div x^{14}\), the bases \(y\) and \(x\) are different, so this rule of subtracting exponents can't be directly applied. Instead, you simply write the expression as a fraction \(\frac{y^{21}}{x^{14}}\), and this becomes its simplified form. Key things to note:

• Only subtract exponents when the bases are the same.
• Keep the base as is when the bases are different.
• If exponents are larger in the numerator for the same base, reduce the exponent in the numerator by the exponent in the denominator.

The division of exponents streamlines problem-solving and helps you simplify expressions accurately while working with variables and constants.