Question

Simplify $$ \frac{y^{6} y^{-3} y^{0.5}}{y^{-2} y^{7}} $$

Short answer

Answer: The simplified form of the expression is \(y^{-1.5}\).

Step-by-step solution

Combine the numerators' exponents

Add the exponents of y in the numerator: $$ y^{6} y^{-3} y^{0.5} = y^{6-3+0.5} = y^{3.5} $$

Combine the denominators' exponents

Add the exponents of y in the denominator: $$ y^{-2} y^{7} = y^{-2+7} = y^5 $$

Simplify the fraction

Divide the entire expression by subtracting the exponents: $$ \frac{y^{3.5}}{y^5} = y^{3.5-5} = y^{-1.5} $$ So the simplified expression is \(y^{-1.5}\).

Key concepts

Understanding Exponents

Exponents are a way of indicating repeated multiplication. For example, in the term \( y^4 \), the exponent "4" tells you to multiply \( y \) by itself 4 times: \( y \times y \times y \times y \). When dealing with expressions involving exponents, there are several important rules to remember:

• Product of Powers Rule: When you multiply terms with the same base, add their exponents: \( y^a \times y^b = y^{a+b} \).
• Quotient of Powers Rule: When you divide terms with the same base, subtract the exponents of the denominator from the numerator: \( \frac{y^a}{y^b} = y^{a-b} \).
• Zero Exponent Rule: Any term raised to the power of zero equals 1: \( y^0 = 1 \). This simplifies calculations but you won't actually multiply by zero.

In the original exercise, you combined the exponents in the numerator and denominator separately using these rules, before simplifying the fraction.

Simplifying Fractions with Exponents

Fraction simplification with exponents involves using the rules of exponents to make expressions more manageable. A fraction is simplified when you have reduced it to its simplest form. In algebra, this often involves simplifying the powers.The process of simplifying a fraction with exponents often includes:

• Adding Exponents in Numerator and Denominator: Combine like terms in the numerator and the denominator. For example, in the exercise, the exponents in the numerator and denominator were summed separately to form simpler expressions.
• Applying the Quotient of Powers Rule: Once you have single terms in the numerator and the denominator, simplify the fraction by subtracting the exponent in the denominator from the exponent in the numerator as was done in the exercise dividing \( y^{3.5} \over y^5 \).

Using these steps can significantly reduce complex fractions to simpler, more understandable expressions.

Basic Algebra Techniques

Algebra is a branch of mathematics that deals with variables and the rules for manipulating these variables. At its core, algebra is used to express and solve equations. In this exercise, algebraic techniques help simplify expressions with variables raised to exponents. Here's how:

• Recognizing Like Terms: In any expression, terms with the same base can be combined, which simplifies calculations.
• Consistent Use of Rules: By consistently applying the exponent rules, one can navigate through seemingly complex expressions and arrive at simpler forms, just like in the given solution.
• Working with Negative Exponents: Negative exponents represent reciprocal values, meaning \( y^{-a} = \frac{1}{y^a} \). This concept is crucial for understanding how expressions transform as they simplify.

These fundamental algebraic techniques help maintain clarity and accuracy in solving and simplifying expressions in mathematics.