Simplify (m^(3/4))/(m^(5/4)m^-2)
The question is asking to perform algebraic simplification on a given expression that involves exponents. Specifically, you need to simplify the expression which consists of m raised to the power of 3/4 divided by m raised to the power of 5/4 multiplied by m raised to the power of -2. The simplification process should use the rules of exponents, which include the quotient rule (when dividing like bases, subtract the exponents), the product rule (when multiplying like bases, add the exponents), and the negative exponent rule (an exponent of -1 means take the reciprocal of the base). The goal is to combine and simplify the expression in terms of m with a single exponent.
$\frac{m^{\frac{3}{4}}}{m^{\frac{5}{4}} m^{- 2}}$
Answer
Solution:
Step:1
Combine the exponents of $m^{\frac{5}{4}}$ and $m^{-2}$ by summing them up.
Step:1.1
Apply the exponent rule $a^{m} \cdot a^{n} = a^{m + n}$ to merge the exponents: $\frac{m^{\frac{3}{4}}}{m^{\frac{5}{4} - 2}}$
Step:1.2
Convert $-2$ into a fraction with the same denominator by multiplying by $\frac{4}{4}$: $\frac{m^{\frac{3}{4}}}{m^{\frac{5}{4} - 2 \cdot \frac{4}{4}}}$
Step:1.3
Combine $-2$ and $\frac{4}{4}$: $\frac{m^{\frac{3}{4}}}{m^{\frac{5}{4} + \frac{-2 \cdot 4}{4}}}$
Step:1.4
Add the numerators over the same denominator: $\frac{m^{\frac{3}{4}}}{m^{\frac{5 - 2 \cdot 4}{4}}}$
Step:1.5
Simplify the expression in the denominator.
Step:1.5.1
Calculate $-2$ times $4$: $\frac{m^{\frac{3}{4}}}{m^{\frac{5 - 8}{4}}}$
Step:1.5.2
Subtract $8$ from $5$: $\frac{m^{\frac{3}{4}}}{m^{\frac{-3}{4}}}$
Step:1.6
Adjust the negative exponent to the front of the fraction: $\frac{m^{\frac{3}{4}}}{m^{-\frac{3}{4}}}$
Step:2
Move $m^{-\frac{3}{4}}$ to the numerator using the rule that $\frac{1}{b^{-n}} = b^{n}$: $m^{\frac{3}{4}} \cdot m^{\frac{3}{4}}$
Step:3
Add the exponents of $m^{\frac{3}{4}}$ and $m^{\frac{3}{4}}$ together.
Step:3.1
Utilize the exponent rule $a^{m} \cdot a^{n} = a^{m + n}$ to combine exponents: $m^{\frac{3}{4} + \frac{3}{4}}$
Step:3.2
Sum the numerators over the common denominator: $m^{\frac{3 + 3}{4}}$
Step:3.3
Add $3$ to $3$: $m^{\frac{6}{4}}$
Step:3.4
Reduce the fraction by eliminating common factors.
Step:3.4.1
Extract a factor of $2$ from $6$: $m^{\frac{2(3)}{4}}$
Step:3.4.2
Eliminate common factors.
Step:3.4.2.1
Extract a factor of $2$ from $4$: $m^{\frac{2 \cdot 3}{2 \cdot 2}}$
Step:3.4.2.2
Cancel out the common factors: $m^{\frac{\cancel{2} \cdot 3}{\cancel{2} \cdot 2}}$
Step:3.4.2.3
Simplify the expression: $m^{\frac{3}{2}}$
Knowledge Notes:
Exponent Rules: The power rule for exponents states that when multiplying two expressions with the same base, you can add the exponents: $a^{m} \cdot a^{n} = a^{m + n}$. When dividing, you subtract the exponents: $\frac{a^{m}}{a^{n}} = a^{m - n}$.
Negative Exponents: A negative exponent indicates that the base is on the opposite side of the fraction line. For example, $a^{-n} = \frac{1}{a^{n}}$.
Combining Fractions: When combining fractions with the same denominator, you can add or subtract the numerators while keeping the denominator the same.
Simplifying Fractions: To simplify a fraction, you can cancel out common factors in the numerator and the denominator.
Common Denominator: When working with fractions that have different denominators, it is often necessary to find a common denominator to combine them. This is typically done by finding the least common multiple of the denominators.
Final Expression: After applying the exponent rules and simplifying, the final expression should be in its simplest form, with no common factors in the numerator and denominator and no negative exponents.
