Simplify the following expressions: 2¹² × 16 × (2³)² ÷ 2⁹, y¹⁰ × y⁵ ÷ y⁷, and ((2²)² × 4) / (8³)

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems:

:

2^{12} \times 16 \times (2^3)^2 \div 2^9

Express all terms with the same base, which is 2.
$16 = 2^4$ $(2^3)^2 = 2^{3 \times 2} = 2^6$
The expression becomes:
$2^{12} \times 2^4 \times 2^6 \div 2^9$
Apply the exponent rules for multiplication ( $a^m \times a^n = a^{m+n}$ ) and division ( $a^m \div a^n = a^{m-n}$ ).
$2^{12+4+6-9}$
Calculate the exponent.
$2^{16+6-9} = 2^{22-9} = 2^{13}$
The final answer is:
$\boxed{2^{13}}$
:
$y^{10} \times y^5 \div y^7$
Apply the exponent rule for multiplication ( $a^m \times a^n = a^{m+n}$ ).
$y^{10+5} \div y^7 = y^{15} \div y^7$
Apply the exponent rule for division ( $a^m \div a^n = a^{m-n}$ ).
$y^{15-7} = y^8$
The final answer is:
$\boxed{y^8}$
:
$\frac{(2^2)^2 \times 4}{(8)^3}$
Express all terms with the same base, which is 2.
$(2^2)^2 = 2^{2 \times 2} = 2^4$ $4 = 2^2$ $8 = 2^3 \implies (8)^3 = (2^3)^3 = 2^{3 \times 3} = 2^9$
The expression becomes:
$\frac{2^4 \times 2^2}{2^9}$
Apply the exponent rule for multiplication in the numerator ( $a^m \times a^n = a^{m+n}$ ).
$\frac{2^{4+2}}{2^9} = \frac{2^6}{2^9}$
Apply the exponent rule for division ( $a^m \div a^n = a^{m-n}$ ).
$2^{6-9} = 2^{-3}$
Express the answer with a positive exponent.
$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
The final answer is:
$\boxed{\frac{1}{8}}$

Step-by-step solution