Simplify (a^9b^5)/(a^3b^15)
The problem is asking for the simplification of an algebraic expression that involves exponents. Specifically, the expression is a fraction where both the numerator and the denominator contain powers of the variables a and b. The process will involve applying the laws of exponents, such as the quotient rule, which states that when dividing like bases, you subtract the exponent in the denominator from the exponent in the numerator. The problem wants the expression to be reduced to its simplest form.
$\frac{a^{9} b^{5}}{a^{3} b^{15}}$
Answer
Solution:
Step 1: Simplify the $a$ terms.
Step 1.1: Extract $a^3$ from the numerator.
$\frac{a^{3}(a^{6}b^{5})}{a^{3}b^{15}}$Step 1.2: Eliminate the common $a^3$ factors.
- Step 1.2.1: Factor out $a^3$ from the denominator.
$\frac{a^{3}(a^{6}b^{5})}{a^{3}(b^{15})}$ - Step 1.2.2: Remove the shared $a^3$ term.
$\frac{\cancel{a^{3}}(a^{6}b^{5})}{\cancel{a^{3}}b^{15}}$ - Step 1.2.3: Present the simplified expression.
$\frac{a^{6}b^{5}}{b^{15}}$
- Step 1.2.1: Factor out $a^3$ from the denominator.
Step 2: Simplify the $b$ terms.
Step 2.1: Extract $b^5$ from the numerator.
$\frac{a^{6}b^{5}}{b^{15}}$Step 2.2: Eliminate the common $b^5$ factors.
- Step 2.2.1: Factor out $b^5$ from the denominator.
$\frac{a^{6}b^{5}}{b^{5}b^{10}}$ - Step 2.2.2: Remove the shared $b^5$ term.
$\frac{\cancel{b^{5}}a^{6}}{\cancel{b^{5}}b^{10}}$ - Step 2.2.3: Present the simplified expression.
$\frac{a^{6}}{b^{10}}$
- Step 2.2.1: Factor out $b^5$ from the denominator.
Knowledge Notes:
The problem involves simplifying an algebraic expression with exponents. The process uses the properties of exponents, specifically the quotient rule, which states that for any non-zero number $a$ and any integers $m$ and $n$, $a^m / a^n = a^{m-n}$, provided that $a$ is not zero.
Here are the relevant knowledge points:
Quotient Rule of Exponents: When dividing like bases, subtract the exponents.
Cancellation: When a factor appears both in the numerator and the denominator, it can be cancelled out.
Simplifying Expressions: The goal is to write the expression in the simplest form by eliminating common factors.
Negative Exponents: If after simplification, an exponent is negative, it indicates that the base is on the opposite side of the fraction (numerator becomes denominator or vice versa).
Writing in Latex: Mathematical expressions are rendered in Latex to provide clear and professional-looking formulas.
In this problem, we first cancel out the common $a^3$ factor, which simplifies to $a^{9-3}$ in the numerator. Then we cancel out the common $b^5$ factor, which simplifies to $b^{5-15}$ in the denominator, resulting in a negative exponent. The final expression is $\frac{a^{6}}{b^{10}}$, which cannot be simplified further because there are no more common factors.
