Rational Exponents - Definition & Examples
Read this topicExplanations 3
- Fraction exponents are like an exponent and a radical Fraction exponents tell you to take the power of a number, and then take a root. It's hard to explain in words, so here's a…
By Zora Gilbert, Joshua Siktar, Jonathan Heller, Caroline K
- A rational exponent can always be turned into a radical. If bbb to the nthn^{th}nth root of a number is real, and mmm is a positive integer, then rational exponents and radicals…
By Ben Ferrara, Moor Xu, Caroline K, Alex Federspiel, Jonathan Heller, Anusha Rahman
- (Video) Simplifying Radical Expressions by mathman1024 This video by mathman1024 works through converting fractional exponents into radicals. Summary\large{\text{Summary}}Summar…
By Alex Federspiel, Amy Feraco, Jonathan Heller, Caroline K, Anusha Rahman
Problems 80
- If y>0y>0y>0, what is y−2/5y^{-2/5}y−2/5 in radical form (or simplified, if possible)?
- If t>0t>0t>0, what is t42t^{\frac42}t24 in radical form (or simplified if possible)?
- What is 8698^{\frac69}896 in radical form (or simplified if possible)?
- If w,z>0w,z>0w,z>0, what is w⋅z34w\cdot z^{\frac34}w⋅z43 in radical form (or simplified if possible)?
- What is 18−1218^{-\frac12}18−21 in radical form (or simplified if possible)?
- If p>0p>0p>0, what is 5(p3)(−53)5(p^3)^{(-\frac53)}5(p3)(−35) in radical form (or simplified if possible)?
- If r,s>0r,s>0r,s>0, what is (r⋅s2)−82(r\cdot s^2)^{-\frac82}(r⋅s2)−28 in radical form (or simplified if possible)?
- What is −y 7-\sqrt{y}^{\,7}−y7 in exponential form, if y>0y>0y>0?
- If t>0t>0t>0, what is 1t5 2\frac{1}{\sqrt[5]{t}^{\,2}}5t21 in exponential form?
- What is (w⋅z5)7(w\cdot \sqrt[5]{z})^7(w⋅5z)7 in exponential form?
- What is 27q3 4\sqrt[3]{27q}^{\,4}327q4 in exponential form?
- In their simplest form, which expressions below have fractional exponents? (Assume all variables below are positive.)
- In their simplest form, which expressions below have fractional exponents? (Assume all variables below are positive.)
- True or False\text{True or False}True or False === Having a rational exponent means that there is a fraction in the exponent.
- True or False\text{True or False}True or False === To evaluate a rational exponent aba^bab where bbb is negative, you flip the fraction in the exponent.
- What is 271327^\frac{1}{3}2731 in its simplest form?
- What is (−8)23(-8)^\frac{2}{3}(−8)32 in its simplest form?
- What is (16)−32(16)^{-\frac{3}{2}}(16)−23 in its simplest form?
- Which are equivalent to 84/38^{4/3}84/3?
- Which are equivalent to 273\sqrt[3]{27}327?
- True or False\text{True or False}True or False === Any number with a rational exponent will simplify to an integer.
- Evaluate 12523.125^{\frac23}.12532.
- Evaluate 4724^{\frac72}427
- Evaluate 20485/11.2048^{5/11}.20485/11.
- Evaluate 8164.81^{\frac64}.8146.
- We can rewrite 23⋅25=2k\sqrt{\sqrt[3]2 \cdot \sqrt[5]2} = 2^k32⋅52=2k for some fractional exponent kkk. Find kkk as a fraction in simplest form.
- If a≠0a\ne0a=0, write aa57\frac{a}{\sqrt[7]{a^5}}7a5a in exponential form.
- Evaluate 9329^\frac{3}{2}923.
- Evaluate 8238^\frac{2}{3}832.
- Evaluate 813481^{\frac{3}{4}}8143.
- Evaluate 811281^\frac{1}{2}8121.
- Evaluate 4124^\frac{1}{2}421.
- Evaluate 2682^\frac{6}{8}286.
- What is another way to write (x)(1b)a(x)^{(\frac{1}{b})a}(x)(b1)a?
- True or False\text{True or False}True or False === bnm=bmn\sqrt[m]{b^n}=b^{\frac{m}{n}}mbn=bnm, if b>1b>1b>1 and m,nm,nm,n are different positive integers.
- True or False\text{True or False}True or False === bmn=bmn\sqrt[n]{b^m}=b^{\frac{m}{n}}nbm=bnm, if b>0b>0b>0 and m,nm,nm,n are positive integers.
- Which expression below is equivalent to bn\sqrt[n]{b}nb?
- Which expression below is equivalent to abn\sqrt[n]{\frac ab}nba?
- Which expression below is equivalent to bn m\sqrt[n]{b}^{\,m}nbm?
- Which expression below is equivalent to (bn)z\sqrt[z]{(\sqrt[n]{b})}z(nb)?
- When evaluating rational exponents:
- Which expression below is equivalent to 4124^\frac{1}{2}421?
- Which expression below is equivalent to 9129^\frac{1}{2}921?
- Evaluate 9129^\frac{1}{2}921.
- Evaluate 4324^\frac{3}{2}423.
- Evaluate 4424^\frac{4}{2}424.
- Which is equivalent to 3423^\frac{4}{2}324?
- If t>0t>0t>0, what is t(12)t^{(\frac12)}t(21) in radical form (or simplified if possible)?
- If t>0t>0t>0, what is t32t^{\frac32}t23 in radical form (or simplified if possible)?
- If t>0t>0t>0, what is t62t^{\frac62}t26 in radical form (or simplified if possible)?
- When raising a quantity to the power ab,\frac{a}{b},ba, where ab\frac{a}{b}ba is a rational number, how could you proceed? Select all possible methods.
- anm=?a^{\frac{n}{m}} = ?amn=? (Assume a,n,ma,n,ma,n,m are all positive.)
- If t>0t>0t>0, what is t65t^{\frac65}t56 in radical form (or simplified if possible)?
- If t>0t>0t>0, what is t34t^{\frac34}t43 in radical form (or simplified if possible)?
- If t>0t>0t>0, what is t(1/4)t^{(1/4)}t(1/4) in radical form (or simplified if possible)?
- If t>0t>0t>0, what is t1nt^{\frac{1}{n}}tn1 in radical form?
- To help understand how to convert from radical to exponential form, evaluate the following. t3\sqrt[3]{t}3t
- Evaluate: t3 2\sqrt[3]{t}^{\,2}3t2
- Evaluate: t2 3\sqrt[2]{t}^{\,3}2t3
- Evaluate: t3 4\sqrt[3]{t}^{\,4}3t4
- Which of the following equals t3 4\sqrt[3]{t}^{\,4}3t4?
- Rewrite t3\sqrt[3]{t}3t in exponential form.
- Evaluate: t3 6\sqrt[3]{t}^{\,6}3t6
- If t>0t>0t>0, what is 1t\frac{1}{\sqrt{t}}t1 in exponential form?
- If t>0t>0t>0, what is 1t3 2\frac{1}{\sqrt[3]{t}^{\,2}}3t21 in exponential form?
- If t>0t>0t>0, what is 6t2 3\frac{6}{\sqrt[2]{t}^{\,3}}2t36 in exponential form?
- Which of the following operations, when applied to only rational numbers, could cause it to become irrational?
- Evaluate the following expression: 493249^{\frac32}4923
- Evaluate the following expression: 271327^{\frac13}2731
- Evaluate the following expression: 21623216^{\frac23}21632
- Evaluate the following expression: 132213^{\frac22}1322
- What is 363236^{\frac32}3623?
- What is (−2)23(-2)^{\frac23}(−2)32
- Which of the following are equal to 8238^{\frac{2}{3}}832?
- What is 163416^{\frac34}1643?
- aaa and bbb are two whole numbers. If 2ab=22^{\frac{a}{b}} = 22ba=2, what can we conclude about aaa and bbb?
- What is 8−238^{-\frac23}8−32?
- What is 161216^{\frac{1}{2}}1621?
- Which expression is equivalent to x12x^{\frac12}x21?
- What is the value of aaa?