Simplify: $(2^{18} \div 2^{12}) \times 2^{4}$
Simplify: $(5^{-4} \div 5^{-9}) \times 5^{-2}$
Simplify: $\frac{(3^{2})^{4} \times 3^{5}}{3^{6}}$
Simplify: $\frac{10^{3} \times 5^{2} \times 2^{4}}{5 \times 2^{3}}$
Simplify: $4^{3} \times \left(\frac{3}{4}\right)^{3}$
Simplify: $(3^{0} + 2^{-2}) \times 4^{2}$
Simplify: $(2^{-3} \times 4^{-2}) \div 2^{-4}$
Find the value of: $\left(\frac{1}{3}\right)^{-2} + \left(\frac{1}{5}\right)^{-2}$
Simplify: $(-1)^{6} + (-1)^{9} + (-1)^{4}$
Find the value of: $(-3)^{4} + (-3)^{2}$
Solve for $m$: If $7^{2m-1} = \frac{1}{49}$, find $m$
To solve these, we use a few fundamental laws:
Product Rule: $a^m \times a^n = a^{m+n}$
Quotient Rule: $a^m \div a^n = a^{m-n}$
Power of a Power: $(a^m)^n = a^{m \times n}$
Negative Exponents: $a^{-n} = \frac{1}{a^n}$
Zero Power: $a^0 = 1$
$(2^{18} \div 2^{12}) \times 2^{4}$
Step 1: Inside the parentheses, subtract the exponents: $2^{18-12} = 2^6$.
Step 2: Multiply by $2^4$ by adding the exponents: $2^{6+4} = 2^{10}$.
Result: $1024$ (or $2^{10}$).
$(5^{-4} \div 5^{-9}) \times 5^{-2}$
Step 1: Subtract the exponents: $-4 – (-9) = -4 + 9 = 5$. So, $5^5$.
Step 2: Multiply by $5^{-2}$: $5^{5 + (-2)} = 5^3$.
Result: $125$.
$\frac{(3^{2})^{4} \times 3^{5}}{3^{6}}$
Step 1: Simplify the power of a power: $3^{2 \times 4} = 3^8$.
Step 2: Combine the top: $3^8 \times 3^5 = 3^{13}$.
Step 3: Divide by the bottom: $3^{13-6} = 3^7$.
Result: $2187$.
$\frac{10^{3} \times 5^{2} \times 2^{4}}{5 \times 2^{3}}$
Step 1: Break $10^3$ into $(2 \times 5)^3$, which is $2^3 \times 5^3$.
Step 2: Combine the top: $(2^3 \times 2^4) \times (5^3 \times 5^2) = 2^7 \times 5^5$.
Step 3: Divide: $2^{7-3} \times 5^{5-1} = 2^4 \times 5^4$.
Result: $10,000$.
$4^{3} \times \left(\frac{3}{4}\right)^{3}$
Step 1: Apply the power to the fraction: $\frac{3^3}{4^3}$.
Step 2: Notice the $4^3$ in the numerator and denominator cancels out.
Result: $3^3 =$ $27$.
$(3^{0} + 2^{-2}) \times 4^{2}$
Step 1: $3^0 = 1$ and $2^{-2} = \frac{1}{4}$.
Step 2: $(1 + 0.25) \times 16 = 1.25 \times 16$.
Result: $20$.
$(2^{-3} \times 4^{-2}) \div 2^{-4}$
Step 1: Convert $4$ to base $2$: $4^{-2} = (2^2)^{-2} = 2^{-4}$.
Step 2: Combine terms: $2^{-3} \times 2^{-4} = 2^{-7}$.
Step 3: Divide: $2^{-7} \div 2^{-4} = 2^{-7 – (-4)} = 2^{-3}$.
Result: $\frac{1}{8}$.
$\left(\frac{1}{3}\right)^{-2} + \left(\frac{1}{5}\right)^{-2}$
Step 1: A negative exponent flips the fraction: $3^2 + 5^2$.
Step 2: $9 + 25 = 34$.
Result: $34$.
$(-1)^{6} + (-1)^{9} + (-1)^{4}$
Step 1: Even powers of $-1$ are $1$; odd powers are $-1$.
Step 2: $1 + (-1) + 1$.
Result: $1$.
$(-3)^{4} + (-3)^{2}$
Step 1: $(-3)^4 = 81$ (even power, positive result).
Step 2: $(-3)^2 = 9$.
Result: $81 + 9 =$ $90$.
$7^{2m-1} = \frac{1}{49}$
Step 1: Rewrite $\frac{1}{49}$ as $7^{-2}$.
Step 2: Set the exponents equal: $2m – 1 = -2$.
Step 3: Solve: $2m = -1 \rightarrow m = -0.5$.
Result: $m = -\frac{1}{2}$.