Laws of Exponents — Dynamic Exam

\(8^{5} \cdot 8^{4}\)

\(a^n \cdot a^m = a^{n+m}\)

When multiplying powers with the same base, add the exponents: \(5 + 4 = 9\)

\(8^{5} \cdot 8^{4} = 8^{9}\)

\(10^{2} \cdot 10^{4}\)

\(a^n \cdot a^m = a^{n+m}\)

Add the exponents: \(2 + 4 = 6\)

\(10^{2} \cdot 10^{4} = 10^{6}\)

\(m^{10} \cdot m^{8}\)

\(a^n \cdot a^m = a^{n+m}\)

Add the exponents: \(10 + 8 = 18\)

\(m^{10} \cdot m^{8} = m^{18}\)

\(b^{8} \cdot b^{8}\)

\(a^n \cdot a^m = a^{n+m}\)

Add the exponents: \(8 + 8 = 16\)

\(b^{8} \cdot b^{8} = b^{16}\)

\(3^{2} \cdot 3^{5} \cdot 3^{6}\)

\(a^n \cdot a^m = a^{n+m}\)

Add all the exponents: \(2 + 5 + 6 = 13\)

\(3^{2} \cdot 3^{5} \cdot 3^{6} = 3^{13}\)

\(b \cdot b^{4}\)

\(a^n \cdot a^m = a^{n+m}\)

The exponent of b is 1, so we add: \(1 + 4 = 5\)

\(b \cdot b^{4} = b^{5}\)

\(4x^{8} \cdot 7x^{10}\)

\(a^n \cdot a^m = a^{n+m}\)

Multiply the coefficients: \(4 \times 7 = 28\), and add the exponents: \(8 + 10 = 18\)

\(4x^{8} \cdot 7x^{10} = 28x^{18}\)

\(13a^{7} \cdot 4a^{2}\)

\(a^n \cdot a^m = a^{n+m}\)

Multiply the coefficients: \(13 \times 4 = 52\), and add the exponents: \(7 + 2 = 9\)

\(13a^{7} \cdot 4a^{2} = 52a^{9}\)

\(x^{5}y^{5} \cdot x^{6}y^{10}\)

\(a^n \cdot a^m = a^{n+m}\)

Add the exponents for each variable: \(x: 5+6=11\), \(y: 5+10=15\)

\(x^{5}y^{5} \cdot x^{6}y^{10} = x^{11}y^{15}\)

\(4^{9} \cdot 4^{12}\)

\(a^n \cdot a^m = a^{n+m}\)

Add the exponents: \(9 + 12 = 21\)

\(4^{9} \cdot 4^{12} = 4^{21}\)

\(5m^{5} \cdot 2m^{7} \cdot 6m\)

\(a^n \cdot a^m = a^{n+m}\)

Multiply the coefficients: \(5 \times 2 \times 6=60\), add the exponents: \(5+7+1=13\)

\(5m^{5} \cdot 2m^{7} \cdot 6m = 60m^{13}\)

\((-2)^{2} \cdot (-2)^{4}\)

\(a^n \cdot a^m = a^{n+m}\)

Even with a negative base, add the exponents: \(2 + 4 = 6\)

\((-2)^{2} \cdot (-2)^{4} = (-2)^{6}\)

\(\frac{4^{8}}{4^{4}}\)

\(\frac{a^n}{a^m} = a^{n-m}\)

When dividing powers with the same base, subtract the exponents: \(8 - 4 = 4\)

\(\frac{4^{8}}{4^{4}} = 4^{4}\)

\(\frac{3^{12}}{3^{8}}\)

\(\frac{a^n}{a^m} = a^{n-m}\)

Subtract the exponents: \(12 - 8 = 4\)

\(\frac{3^{12}}{3^{8}} = 3^{4}\)

\(\frac{y^{24}}{y^{14}}\)

\(\frac{a^n}{a^m} = a^{n-m}\)

Subtract the exponents: \(24 - 14 = 10\)

\(\frac{y^{24}}{y^{14}} = y^{10}\)

\(\frac{a^{15}}{a^{8}}\)

\(\frac{a^n}{a^m} = a^{n-m}\)

Subtract the exponents: \(15 - 8 = 7\)

\(\frac{a^{15}}{a^{8}} = a^{7}\)

\(\frac{20y^{11}}{4y^{5}}\)

\(\frac{a^n}{a^m} = a^{n-m}\)

Divide the coefficients: \(20 \div 4=5\), subtract the exponents: \(11-5=6\)

\(\frac{20y^{11}}{4y^{5}} = 5y^{6}\)

\(\frac{36x^{10}}{9x^{3}}\)

\(\frac{a^n}{a^m} = a^{n-m}\)

Divide the coefficients: \(36 \div 9=4\), subtract the exponents: \(10-3=7\)

\(\frac{36x^{10}}{9x^{3}} = 4x^{7}\)

\(\frac{a^{8}b^{12}}{a^{3}b^{7}}\)

\(\frac{a^n}{a^m} = a^{n-m}\)

Subtract the exponents for each variable: \(a: 8-3=5\), \(b: 12-7=5\)

\(\frac{a^{8}b^{12}}{a^{3}b^{7}} = a^{5}b^{5}\)

\(\frac{x^{7} \cdot x^{5}}{x^{3}}\)

\(\frac{a^n}{a^m} = a^{n-m}\)

First add the top exponents: \(7+5=12\), then subtract: \(12-3=9\)

\(\frac{x^{7} \cdot x^{5}}{x^{3}} = x^{9}\)

\(\frac{2^{15}}{2^{10}}\)

\(\frac{a^n}{a^m} = a^{n-m}\)

Subtract the exponents: \(15 - 10 = 5\)

\(\frac{2^{15}}{2^{10}} = 2^{5}\)

\(\frac{m^{10} \cdot m^{4}}{m^{6}}\)

\(\frac{a^n}{a^m} = a^{n-m}\)

First add the top exponents: \(10+4=14\), then subtract: \(14-6=8\)

\(\frac{m^{10} \cdot m^{4}}{m^{6}} = m^{8}\)

\(\frac{50a^{9}b^{6}}{10a^{4}b^{2}}\)

\(\frac{a^n}{a^m} = a^{n-m}\)

Divide the coefficients: \(50 \div 10=5\), subtract the exponents: \(a:9-4=5\), \(b:6-2=4\)

\(\frac{50a^{9}b^{6}}{10a^{4}b^{2}} = 5a^{5}b^{4}\)

\(\frac{y^n}{y^{3}}\)

\(\frac{a^n}{a^m} = a^{n-m}\)

Subtract the exponents: \(n - 3 = n-3\)

\(\frac{y^n}{y^{3}} = y^{n-3}\)

\((3^{2})^{3}\)

\((a^n)^m = a^{n \cdot m}\)

For a power of a power, multiply the exponents: \(2 \times 3 = 6\)

\((3^{2})^{3} = 3^{6}\)

\((3^{4})^{2}\)

\((a^n)^m = a^{n \cdot m}\)

Multiply the exponents: \(4 \times 2 = 8\)

\((3^{4})^{2} = 3^{8}\)

\((x^{5})^{4}\)

\((a^n)^m = a^{n \cdot m}\)

Multiply the exponents: \(5 \times 4 = 20\)

\((x^{5})^{4} = x^{20}\)

\((a^{3})^{5}\)

\((a^n)^m = a^{n \cdot m}\)

Multiply the exponents: \(3 \times 5 = 15\)

\((a^{3})^{5} = a^{15}\)

\((2x^{3})^{4}\)

\((a^n)^m = a^{n \cdot m}\)

The coefficient is also raised to the power: \(2^{4}=16\), and the exponents are multiplied: \(3 \times 4=12\)

\((2x^{3})^{4} = 16x^{12}\)

\((5y^{2})^{3}\)

\((a^n)^m = a^{n \cdot m}\)

The coefficient is raised to the power: \(5^{3}=125\), exponents are multiplied: \(2 \times 3=6\)

\((5y^{2})^{3} = 125y^{6}\)

\((2^{2})^{3} \cdot 2^{5}\)

\((a^n)^m = a^{n \cdot m}\)

First, the power of a power: \(2 \times 3=6\), then multiply powers: \(6+5=11\)

\((2^{2})^{3} \cdot 2^{5} = 2^{11}\)

\((x^{5})^{2} \cdot (x^{3})^{2}\)

\((a^n)^m = a^{n \cdot m}\)

First multiply the exponents: \(5 \times 2=10\), \(3 \times 2=6\), then add: \(10+6=16\)

\((x^{5})^{2} \cdot (x^{3})^{2} = x^{16}\)

\(((a^{2})^{3})^{4}\)

\((a^n)^m = a^{n \cdot m}\)

Multiply all the exponents: \(2 \times 3 \times 4 = 24\)

\(((a^{2})^{3})^{4} = a^{24}\)

\((3a^{4}b^{2})^{3}\)

\((a^n)^m = a^{n \cdot m}\)

Coefficient: \(3^{3}=27\), exponents: \(a: 4 \times 3=12\), \(b: 2 \times 3=6\)

\((3a^{4}b^{2})^{3} = 27a^{12}b^{6}\)

\((10^{3})^{2}\)

\((a^n)^m = a^{n \cdot m}\)

Multiply the exponents: \(3 \times 2 = 6\)

\((10^{3})^{2} = 10^{6}\)

\(\frac{(x^{6})^{3}}{x^{8}}\)

\((a^n)^m = a^{n \cdot m}\)

First, the power of a power: \(6 \times 3=18\), then divide: \(18-8=10\)

\(\frac{(x^{6})^{3}}{x^{8}} = x^{10}\)

\((3 \cdot 4)^{2}\)

\((a \cdot b)^n = a^n \cdot b^n\)

Expand: \(3^{2} \cdot 4^{2} = 9 \cdot 16 = 144\)

\((3 \cdot 4)^{2} = 144\)

\((2 \cdot 5)^{4}\)

\((a \cdot b)^n = a^n \cdot b^n\)

You can compute: \((2 \cdot 5)^{4} = 10^{4} = 10000\)

\((2 \cdot 5)^{4} = 10000\)

\((2xy)^{3}\)

\((a \cdot b)^n = a^n \cdot b^n\)

Each factor is raised to the power: \(2^{3}=8\), \(x^{3}\), \(y^{3}\)

\((2xy)^{3} = 8x^{3}y^{3}\)

\((5ab)^{2}\)

\((a \cdot b)^n = a^n \cdot b^n\)

Each factor is raised to the power: \(5^{2}=25\)

\((5ab)^{2} = 25a^{2}b^{2}\)

\(x^{4} \cdot y^{4}\)

\((a \cdot b)^n = a^n \cdot b^n\)

When the exponent is the same, you can combine: \(x^{4} \cdot y^{4} = (xy)^{4}\)

\(x^{4} \cdot y^{4} = (xy)^{4}\)

\(2^{5} \cdot 3^{5}\)

\((a \cdot b)^n = a^n \cdot b^n\)

When the exponent is the same: \(2^{5} \cdot 3^{5} = (2 \cdot 3)^{5} = 6^{5}\)

\(2^{5} \cdot 3^{5} = 6^{5}\)

\((x^{2}y^{3})^{4}\)

\((a \cdot b)^n = a^n \cdot b^n\)

Each exponent is multiplied by 4: \(x: 2 \times 4=8\), \(y: 3 \times 4=12\)

\((x^{2}y^{3})^{4} = x^{8}y^{12}\)

\((3x^{3})^{3}\)

\((a \cdot b)^n = a^n \cdot b^n\)

Coefficient: \(3^{3}=27\), exponent: \(3 \times 3=9\)

\((3x^{3})^{3} = 27x^{9}\)

\((2a^{2}b)^{5}\)

\((a \cdot b)^n = a^n \cdot b^n\)

Coefficient: \(2^{5}=32\), \(a: 2 \times 5=10\), \(b: 1 \times 5=5\)

\((2a^{2}b)^{5} = 32a^{10}b^{5}\)

\((-2x)^{4}\)

\((a \cdot b)^n = a^n \cdot b^n\)

Even exponent: \((-2)^{4} = 16\) (positive because 4 is even)

\((-2x)^{4} = 16x^{4}\)

\((xy^{2}z^{3})^{2}\)

\((a \cdot b)^n = a^n \cdot b^n\)

Each exponent is multiplied by 2: \(x: 1 \times 2=2\), \(y: 2 \times 2=4\), \(z: 3 \times 2=6\)

\((xy^{2}z^{3})^{2} = x^{2}y^{4}z^{6}\)

\((4a^3b^2)^2 \cdot (2ab)^3\)

\((a \cdot b)^n = a^n \cdot b^n\)

First expand each set of parentheses: \((4a^3b^2)^2=16a^6b^4\), \((2ab)^3=8a^3b^3\), then multiply: \(16 \cdot 8=128\), \(a^6 \cdot a^3=a^9\), \(b^4 \cdot b^3=b^7\)

\((4a^3b^2)^2 \cdot (2ab)^3 = 128a^9b^7\)

\(\left(\frac{2}{3}\right)^{3}\)

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Raise the numerator and denominator to the power: \(\frac{2^{3}}{3^{3}} = \frac{8}{27}\)

\(\left(\frac{2}{3}\right)^{3} = \frac{8}{27}\)

\(\left(\frac{5}{2}\right)^{2}\)

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Raise the numerator and denominator to the power: \(\frac{5^{2}}{2^{2}} = \frac{25}{4}\)

\(\left(\frac{5}{2}\right)^{2} = \frac{25}{4}\)

\(\left(\frac{x}{y}\right)^{4}\)

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Raise the numerator and denominator to the power 4

\(\left(\frac{x}{y}\right)^{4} = \frac{x^{4}}{y^{4}}\)

\(\left(\frac{a}{b}\right)^{6}\)

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Raise the numerator and denominator to the power 6

\(\left(\frac{a}{b}\right)^{6} = \frac{a^{6}}{b^{6}}\)

\(\left(\frac{x^{3}}{y^{2}}\right)^{4}\)

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Multiply the exponents — numerator: \(3 \times 4=12\), denominator: \(2 \times 4=8\)

\(\left(\frac{x^{3}}{y^{2}}\right)^{4} = \frac{x^{12}}{y^{8}}\)

\(\left(\frac{3a}{b}\right)^{3}\)

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Coefficient: \(3^{3}=27\), a and b are raised to the power 3

\(\left(\frac{3a}{b}\right)^{3} = \frac{27a^{3}}{b^{3}}\)

\(\left(\frac{2x^{2}}{3y}\right)^{2}\)

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Numerator: \(2^{2}=4\), \(x: 2 \times 2=4\). Denominator: \(3^{2}=9\), \(y: 1 \times 2=2\)

\(\left(\frac{2x^{2}}{3y}\right)^{2} = \frac{4x^{4}}{9y^{2}}\)

\(\frac{a^{5}}{b^{5}}\)

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

When the numerator and denominator have the same exponent, you can write it as a power of a fraction

\(\frac{a^{5}}{b^{5}} = \left(\frac{a}{b}\right)^{5}\)

\(\left(\frac{a^{2}b}{c}\right)^{3}\)

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Each exponent is multiplied by 3: \(a: 2 \times 3=6\), \(b: 1 \times 3=3\), \(c: 1 \times 3=3\)

\(\left(\frac{a^{2}b}{c}\right)^{3} = \frac{a^{6}b^{3}}{c^{3}}\)

\(\left(\frac{a^{2}b^{3}}{c^{4}}\right)^{5}\)

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Each exponent is multiplied by 5: \(a: 2 \times 5=10\), \(b: 3 \times 5=15\), \(c: 4 \times 5=20\)

\(\left(\frac{a^{2}b^{3}}{c^{4}}\right)^{5} = \frac{a^{10}b^{15}}{c^{20}}\)

\(\left(\frac{4}{5}\right)^{3}\)

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

Raise the numerator and denominator to the power: \(4^{3}=64\), \(5^{3}=125\)

\(\left(\frac{4}{5}\right)^{3} = \frac{64}{125}\)

\(\left(\frac{x^2}{y}\right)^3 \cdot \left(\frac{x}{y^2}\right)^2\)

\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

First expand: \(\frac{(x^2)^3}{y^3}=\frac{x^6}{y^3}\), \(\frac{x^2}{(y^2)^2}=\frac{x^2}{y^4}\), then multiply: \(\frac{x^6 \cdot x^2}{y^3 \cdot y^4}=\frac{x^8}{y^7}\)

\(\left(\frac{x^2}{y}\right)^3 \cdot \left(\frac{x}{y^2}\right)^2 = \frac{x^8}{y^7}\)