Question

dividing powers w
$a^m \div a^$
power of a power:
$(a^m)^n$
example 1 simplify,
a)
$\frac{3^6 \times 2^5}{2^2 \times 3^4} = 3^{6 - 4} \times $
$= 3^2 \times 2$
b)
$\frac{4p^7 \times 6p^9}{12p^{15}} = \frac{24p}{12p^{15}}$
c)
$\frac{(x^{a + b})(x^{2a - b})}{4x^{a - 2b}} = $

Explanation:

Step1: Simplify the numerator (multiply powers)

When multiplying powers with the same base, we add the exponents: \( (x^{a + b})(x^{2a - b}) = x^{(a + b)+(2a - b)} \).
Simplify the exponent: \( (a + b)+(2a - b)=a + b + 2a - b = 3a \), so the numerator becomes \( x^{3a} \).

Step2: Divide by the denominator (divide powers)

Now we have \( \frac{x^{3a}}{4x^{a - 2b}} \). When dividing powers with the same base, we subtract the exponents: \( \frac{x^{3a}}{x^{a - 2b}}=x^{3a-(a - 2b)} \).
Simplify the exponent: \( 3a-(a - 2b)=3a - a + 2b = 2a + 2b \).

Step3: Combine with the coefficient

The coefficient is \( \frac{1}{4} \), so the simplified form is \( \frac{1}{4}x^{2a + 2b} \) (or we can factor the exponent: \( \frac{1}{4}x^{2(a + b)} \)).

Answer:

\( \frac{1}{4}x^{2a + 2b} \) (or \( \frac{x^{2(a + b)}}{4} \))