Write in the Form ax<sup>p</sup>

Jump to section:

Key Ideas Easy Medium Hard

\(a^{\frac{1}{n}}=\sqrt[n]{a}\)

\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m\)

\(a^{-n}=\dfrac{1}{a^n}\)

What to Remember

A root is a power

The formula is

\(\sqrt[n]{a}=a^{\frac{1}{n}}\)

\(\sqrt{x}=x^{\frac{1}{2}}\)

\(\sqrt[3]{x}=x^{\frac{1}{3}}\)

Product rule

Same base, multiply: add the exponents.

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

\(x^3\sqrt{x}=x^3\cdot x^{\frac{1}{2}}\)

\(x^{\frac{7}{2}}\)

Quotient rule

Same base, divide: subtract the exponents.

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

\(\dfrac{6x}{\sqrt[3]{x}}=6x^{1-\frac{1}{3}}\)

\(6x^{\frac{2}{3}}\)

Two terms

Split a sum or difference over a monomial, then write each piece as a power.

\(\dfrac{A+B}{C}=\dfrac{A}{C}+\dfrac{B}{C}\)

Level 1

Easy

Write each root as a power, then add the exponents.

Example

Write in the form \(ax^p\). \(x^3\sqrt{x}\)

The formula is.\(\sqrt{x}=x^{\frac{1}{2}}\)

\(x^3\sqrt{x}=x^3\cdot x^{\frac{1}{2}}\)

The formula is.\(a^m\cdot a^n=a^{m+n}\)

\(x^{3+\frac{1}{2}}\)

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

\(\dfrac{6}{2}+\dfrac{1}{2}=\dfrac{7}{2}\)

\(x^{\frac{7}{2}}\)

Answer: \(x^{\frac{7}{2}}\)

Write in the form \(ax^p\). \(x\sqrt{x}\)

Show Answer

\(x\sqrt{x}=x^1\cdot x^{\frac{1}{2}}\)

\(x^{1+\frac{1}{2}}\)

\(1=\dfrac{2}{2}\)

\(\dfrac{2}{2}+\dfrac{1}{2}=\dfrac{3}{2}\)

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

Answer: \(x^{\frac{3}{2}}\)

Write in the form \(ax^p\). \(x^4\sqrt[3]{x}\)

Show Answer

\(\sqrt[3]{x}=x^{\frac{1}{3}}\)

\(x^4\cdot x^{\frac{1}{3}}=x^{4+\frac{1}{3}}\)

\(4=\dfrac{12}{3}\)

\(\dfrac{12}{3}+\dfrac{1}{3}=\dfrac{13}{3}\)

\(x^{\frac{13}{3}}\)

Answer: \(x^{\frac{13}{3}}\)

Write in the form \(ax^p\). \(x^2\sqrt[3]{x}\)

Show Answer

\(x^2\cdot x^{\frac{1}{3}}=x^{2+\frac{1}{3}}\)

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

\(\dfrac{6}{3}+\dfrac{1}{3}=\dfrac{7}{3}\)

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

Answer: \(x^{\frac{7}{3}}\)

Write in the form \(ax^p\). \(x^5\sqrt{x}\)

Show Answer

\(x^5\cdot x^{\frac{1}{2}}=x^{5+\frac{1}{2}}\)

\(5=\dfrac{10}{2}\)

\(\dfrac{10}{2}+\dfrac{1}{2}=\dfrac{11}{2}\)

\(x^{\frac{11}{2}}\)

Answer: \(x^{\frac{11}{2}}\)

Write in the form \(ax^p\). \(5x\sqrt[3]{x}\)

Show Answer

\(5x\cdot x^{\frac{1}{3}}=5x^{1+\frac{1}{3}}\)

\(1=\dfrac{3}{3}\)

\(\dfrac{3}{3}+\dfrac{1}{3}=\dfrac{4}{3}\)

\(5x^{\frac{4}{3}}\)

Answer: \(5x^{\frac{4}{3}}\)

Level 2

Medium

Write roots as powers, then subtract the exponents. Keep the coefficient.

Example

Write in the form \(ax^p\). \(\dfrac{6x}{\sqrt[3]{x}}\)

The formula is.\(\sqrt[3]{x}=x^{\frac{1}{3}}\)

\(\dfrac{6x}{\sqrt[3]{x}}=\dfrac{6x}{x^{\frac{1}{3}}}\)

The formula is.\(\dfrac{a^m}{a^n}=a^{m-n}\)

\(6x^{1-\frac{1}{3}}\)

\(1=\dfrac{3}{3}\)

\(\dfrac{3}{3}-\dfrac{1}{3}=\dfrac{2}{3}\)

\(6x^{\frac{2}{3}}\)

Answer: \(6x^{\frac{2}{3}}\)

Write in the form \(ax^p\). \(\dfrac{x^3}{\sqrt{x}}\)

Show Answer

\(\dfrac{x^3}{x^{\frac{1}{2}}}=x^{3-\frac{1}{2}}\)

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

\(\dfrac{6}{2}-\dfrac{1}{2}=\dfrac{5}{2}\)

\(x^{\frac{5}{2}}\)

Answer: \(x^{\frac{5}{2}}\)

Write in the form \(ax^p\). \(\dfrac{2x^2}{\sqrt[3]{x}}\)

Show Answer

\(\dfrac{2x^2}{x^{\frac{1}{3}}}=2x^{2-\frac{1}{3}}\)

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

\(\dfrac{6}{3}-\dfrac{1}{3}=\dfrac{5}{3}\)

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

Answer: \(2x^{\frac{5}{3}}\)

Write in the form \(ax^p\). \(\dfrac{3\sqrt{x}}{x^2}\)

Show Answer

\(\dfrac{3x^{\frac{1}{2}}}{x^2}=3x^{\frac{1}{2}-2}\)

\(2=\dfrac{4}{2}\)

\(\dfrac{1}{2}-\dfrac{4}{2}=-\dfrac{3}{2}\)

\(3x^{-\frac{3}{2}}\)

Answer: \(3x^{-\frac{3}{2}}\)

Write in the form \(ax^p\). \(\dfrac{8x}{\sqrt[3]{x^2}}\)

Show Answer

\(\sqrt[3]{x^2}=x^{\frac{2}{3}}\)

\(\dfrac{8x}{x^{\frac{2}{3}}}=8x^{1-\frac{2}{3}}\)

\(1=\dfrac{3}{3}\)

\(\dfrac{3}{3}-\dfrac{2}{3}=\dfrac{1}{3}\)

\(8x^{\frac{1}{3}}\)

Answer: \(8x^{\frac{1}{3}}\)

Write in the form \(ax^p\). \(\dfrac{1}{5\sqrt{x}}\)

Show Answer

\(\dfrac{1}{5x^{\frac{1}{2}}}=\dfrac{1}{5}\,x^{-\frac{1}{2}}\)

Answer: \(\dfrac{1}{5}x^{-\frac{1}{2}}\)

Level 3

Hard

Two terms, a product of roots, or a power of y written back in x.

Example

Write in the form \(ax^p+bx^q\). \(\dfrac{2}{\sqrt[3]{x}}+5\sqrt{x}\)

The formula is.\(\sqrt[3]{x}=x^{\frac{1}{3}}\)

\(\dfrac{2}{\sqrt[3]{x}}=2x^{-\frac{1}{3}}\)

The formula is.\(\sqrt{x}=x^{\frac{1}{2}}\)

\(5\sqrt{x}=5x^{\frac{1}{2}}\)

\(2x^{-\frac{1}{3}}+5x^{\frac{1}{2}}\)

Answer: \(2x^{-\frac{1}{3}}+5x^{\frac{1}{2}}\)

Write in the form \(ax^p\). \(\sqrt[3]{x}\cdot\sqrt[4]{x^3}\)

Show Answer

\(\sqrt[3]{x}=x^{\frac{1}{3}}\)

\(\sqrt[4]{x^3}=x^{\frac{3}{4}}\)

\(x^{\frac{1}{3}}\cdot x^{\frac{3}{4}}=x^{\frac{1}{3}+\frac{3}{4}}\)

\(\dfrac{1}{3}=\dfrac{4}{12}\)

\(\dfrac{3}{4}=\dfrac{9}{12}\)

\(\dfrac{4}{12}+\dfrac{9}{12}=\dfrac{13}{12}\)

\(x^{\frac{13}{12}}\)

Answer: \(x^{\frac{13}{12}}\)

Write in the form \(ax^p\). \(\left(\dfrac{1}{3\sqrt{x}}\right)^2\)

Show Answer

\(\dfrac{1}{3\sqrt{x}}=\dfrac{1}{3}\,x^{-\frac{1}{2}}\)

The formula is.\((ab)^n=a^n b^n\)

\(\left(\dfrac{1}{3}\,x^{-\frac{1}{2}}\right)^2=\dfrac{1}{9}\,x^{-1}\)

Answer: \(\dfrac{1}{9}x^{-1}\)

Write in the form \(x^a+x^b\). \(\dfrac{x^2+\sqrt{x}}{x}\)

Show Answer

\(\dfrac{x^2}{x}+\dfrac{\sqrt{x}}{x}\)

\(\dfrac{x^2}{x}=x\)

\(\dfrac{x^{\frac{1}{2}}}{x}=x^{\frac{1}{2}-1}\)

\(1=\dfrac{2}{2}\)

\(\dfrac{1}{2}-\dfrac{2}{2}=-\dfrac{1}{2}\)

\(x^{-\frac{1}{2}}\)

\(x+x^{-\frac{1}{2}}\)

Answer: \(x+x^{-\frac{1}{2}}\)

Given that \(y=8\sqrt{x}\), write \(\sqrt[3]{y}\) in the form \(ax^k\).

Show Answer

\(y=8x^{\frac{1}{2}}\)

\(\sqrt[3]{y}=\left(8x^{\frac{1}{2}}\right)^{\frac{1}{3}}\)

The formula is.\((ab)^n=a^n b^n\)

\(8^{\frac{1}{3}}\cdot\left(x^{\frac{1}{2}}\right)^{\frac{1}{3}}\)

\(8^{\frac{1}{3}}=2\)

The formula is.\((a^m)^n=a^{mn}\)

\(x^{\frac{1}{2}\cdot\frac{1}{3}}=x^{\frac{1}{6}}\)

\(2x^{\frac{1}{6}}\)

Answer: \(2x^{\frac{1}{6}}\)

Given that \(y=\dfrac{4}{\sqrt[3]{x}}\), write \(y^3\) in the form \(ax^k\).

Show Answer

\(y=4x^{-\frac{1}{3}}\)

\(y^3=\left(4x^{-\frac{1}{3}}\right)^3\)

\(4^3\cdot x^{-1}\)

\(64x^{-1}\)

Answer: \(64x^{-1}\)

Need help writing expressions as a single power?

Book a Session with Amy