Problem 56 - Algebra A Combined Function

Question

Use the quotient rule and simplify each expression. $$ \frac{y^{10}}{y^{9}} $$

Step-by-Step Solution

Answer

y

Step 1: Apply the Quotient Rule

The quotient rule for exponents states that when you divide like bases, you can subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \). Here, the bases are the same \( y \), so apply the rule: \( \frac{y^{10}}{y^9} = y^{10-9} \).

Step 2: Simplify the Expression

Now, simplify the resulting expression: \( y^{10-9} = y^1 \). Since any number raised to the power of 1 is itself, this simplifies further to \( y \).

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making mathematical expressions as straightforward as possible. In algebra, simplification involves reducing an expression to its simplest form. This usually means performing operations such as addition, subtraction, multiplication, or division, and eliminating any unnecessary terms or factors.

To simplify an expression like \( \frac{y^{10}}{y^9} \), first apply the quotient rule of exponents, which is a technique used to streamline expressions involving powers with the same base. Simplification often results in making computations easier, reading expressions cleaner, and solving equations more efficiently.

By simplifying, you remove complexities, making it easier to understand and solve problems in mathematics. Learn to identify like terms and common factors because recognizing patterns in expressions aids in their simplification.

Exponents

Exponents are a shorthand way to express repeated multiplication of a number by itself. In the expression \( a^m \), "a" is the base and "m" is the exponent. The exponent tells you how many times to multiply the base by itself.

Key points to remember about exponents include:

• When multiplying like bases, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• When dividing like bases, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• An exponent of zero means the result is 1: \( a^0 = 1 \), where \( a eq 0 \).
• Negative exponents indicate a reciprocal: \( a^{-n} = \frac{1}{a^n} \).

Recognizing these properties helps simplify expressions quickly. In our given example, using the quotient rule makes the simplification straightforward, as the exponents directly dictate how the division operates.

Algebraic Fractions

Algebraic fractions are ratios of algebraic expressions, similar to numerical fractions but involving polynomials and other algebraic terms instead of just numbers. Simplifying algebraic fractions requires understanding how to manipulate both the numerators and denominators.

When simplifying algebraic fractions involving exponents, observe the principles of reducing fractions, much like reducing numerical fractions by finding common factors. In \( \frac{y^{10}}{y^9} \), the common base \( y \) permits utilizing the quotient rule of exponents, reducing it to a simpler term \( y \).

Algebraic fractions can also involve more complex polynomials as numerators and denominators. Keep in mind:

• Always factor expressions where possible.
• Check for common factors in both the numerator and the denominator.
• Simplify by cancelling out those common factors.

Mastering algebraic fractions ensures that you can efficiently simplify and solve a variety of expressions that you will encounter in algebra.