Question Simplify $p^{3} \times p^{5}$ , $\left(4 a b^{2}\right)^{3}$ , and $\frac{16 m^{7} n^{3}}{4 m^{3} n}$ .

Studdy Solution

STEP 1

Assumptions(a) We are given $p^{3} \times p^{5}$ and we are to simplify it. (b) We are given $\left(4 a b^{}\right)^{3}$ and we are to simplify it. (c) We are given $\frac{16 m^{7} n^{3}}{4 m^{3} n}$ and we are to simplify it.

STEP 2

(a) To simplify $p^{} \times p^{5}$ , we use the rule of exponents which states that when multiplying two powers with the same base, we add the exponents.
$p^{} \times p^{5} = p^{+5}$

STEP 3

(a) Now, add the exponents to simplify the expression.
$p^{3+5} = p^{8}$

STEP 4

(b) To simplify $\left(4 a b^{2}\right)^{3}$ , we use the rule of exponents which states that when raising a product to a power, we raise each factor to that power.
$\left(4 a b^{2}\right)^{3} =4^{3} a^{3} b^{2 \times3}$

STEP 5

(b) Now, calculate the powers to simplify the expression.
$4^{3} a^{3} b^{2 \times3} =64 a^{3} b^{}$

STEP 6

(c) To simplify $\frac{16 m^{} n^{3}}{4 m^{3} n}$ , we divide the numerical coefficients and subtract the exponents of like terms.
$\frac{16 m^{} n^{3}}{4 m^{3} n} = \frac{16}{4} \times \frac{m^{}}{m^{3}} \times \frac{n^{3}}{n}$

STEP 7

(c) Now, simplify the expression.
$\frac{16}{4} \times \frac{m^{7}}{m^{3}} \times \frac{n^{3}}{n} =4 m^{7-3} n^{3-1}$

STEP 8

(c) Finally, subtract the exponents to simplify the expression.
$4 m^{7-3} n^{3-1} =4 m^{4} n^{2}$ The simplified forms of the given expressions are(a) $p^{8}$ (b) $64 a^{3} b^{6}$ (c) $4 m^{4} n^{2}$

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