Let xx = 8, start superscript, one fifth, end superscript8

Question

Let xx = 8, start superscript, one fifth, end superscript8 
5
1
​
 
 . Rewrite the equation in the previous answer box, replacing 8, start superscript, one fifth, end superscript8 
5
1
​
 
  with xx (substitute).

the equation is \left(8^{\frac{1}{5}}\right)^{5}=8

GPT-4o mini · Answer

The original equation is $$ \left(8^{\frac{1}{5}}\right)^{5}=8. $$ Substituting $ x $ for $ 8^{\frac{1}{5}} $, we have: $$ \left(x\right)^{5} = 8. $$ Therefore, the rewritten equation with the substitution is $$ x^{5} = 8. $$

bluesanta · Answer

Solve for a positive value of xx, expressing your answer with a radical symbol.

GPT-4o mini · Answer

We have the equation $$ x^5 = 8. $$ To solve for $ x $, we can take the fifth root of both sides: $$ x = 8^{\frac{1}{5}}. $$ Since we want to express $ 8 $ with a radical symbol, we can write $ 8 $ as $ 2^3 $. Thus, $$ 8^{\frac{1}{5}} = (2^3)^{\frac{1}{5}} = 2^{\frac{3}{5}}. $$ Expressing this with a radical symbol gives us: $$ x = \sqrt[5]{8} = \sqrt[5]{2^3}. $$ So, the positive value of $ x $ is $$ \sqrt[5]{8}. $$

bluesanta · Answer

no what dose x eaqule in the equation

GPT-4o mini · Answer

To solve for $ x $ in the equation $ x^5 = 8 $: 1. We take the fifth root of both sides: $$ x = 8^{\frac{1}{5}}. $$ 2. Since $ 8 $ can be expressed as $ 2^3 $, we can rewrite this as: $$ x = (2^3)^{\frac{1}{5}} = 2^{\frac{3}{5}}. $$ 3. Now, to express $ 2^{\frac{3}{5}} $ with a radical, we have: $$ x = \sqrt[5]{2^3} = \sqrt[5]{8}. $$ Thus, the positive value of $ x $ is $$ x = \sqrt[5]{8} \approx 2^{0.6}. $$ If you want the exact numerical value, $ x $ is approximately $ 2^{0.6} $ or about $ 1.5157 $. However, in radical form, $ x = \sqrt[5]{8} $.

bluesanta · Answer

Let xx = 12, start superscript, one quarter, end superscript12
4
1

. Rewrite the equation in the previous answer box, replacing 12, start superscript, one quarter, end superscript12
4
1

with xx (substitute).

the equiton is (12^1/4)^4 = 12

GPT-4o mini · Answer

The original equation is $$ \left(12^{\frac{1}{4}}\right)^{4} = 12. $$ Substituting $ x $ for $ 12^{\frac{1}{4}} $, we have: $$ \left(x\right)^{4} = 12. $$ Therefore, the rewritten equation with the substitution is $$ x^{4} = 12. $$

bluesanta · Answer

Let xx = 8, start superscript, one third, end superscript8
3
1

. Rewrite the equation in the previous answer box, replacing 8, start superscript, one third, end superscript8
3
1

with xx (substitute).
the equation is \left(8^{\frac{1}{3}}\right)^{3}=8

GPT-4o mini · Answer

The original equation is $$ \left(8^{\frac{1}{3}}\right)^{3} = 8. $$ Substituting $ x $ for $ 8^{\frac{1}{3}} $, we have: $$ \left(x\right)^{3} = 8. $$ Therefore, the rewritten equation with the substitution is $$ x^{3} = 8. $$

bluesanta · Answer

Evaluate the left hand side to find the value of a,a in the equation in simplest form.
x, start superscript, start fraction, 3, divided by, 5, end fraction, end superscript, x, start superscript, one sixth, end superscript, equals, x, to the power a
x
5
3

x
6
1

=
x
a

GPT-4o mini · Answer

To evaluate the left-hand side of the equation

\[ x^{\frac{3}{5}} \cdot x^{\frac{1}{6}} = x^a, \]

we can use the property of exponents that states $ x^m \cdot x^n = x^{m+n} $.

First, we need to add the exponents $ \frac{3}{5} $ and $ \frac{1}{6} $:

\[ a = \frac{3}{5} + \frac{1}{6}. \]
To add these fractions, we need a common denominator. The least common multiple of $ 5 $ and $ 6 $ is $ 30 $.
- Convert $ \frac{3}{5} $ to have a denominator of $ 30 $:
\[ \frac{3}{5} = \frac{3 \cdot 6}{5 \cdot 6} = \frac{18}{30}. \]
- Convert $ \frac{1}{6} $ to have a denominator of $ 30 $:
\[ \frac{1}{6} = \frac{1 \cdot 5}{6 \cdot 5} = \frac{5}{30}. \]
Now we can add the two fractions:

\[ a = \frac{18}{30} + \frac{5}{30} = \frac{18 + 5}{30} = \frac{23}{30}. \]

Thus, the value of $ a $ in simplest form is

\[ \boxed{\frac{23}{30}}. \]