Evaluate the left hand side to find the value of a,a in the equation in simplest form.

Question

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

Answer 1

To evaluate the left-hand side of the equation \( x^{\frac{3}{5}} \cdot x^{\frac{1}{6}} \) and equate it to \( x^a \), we can use the properties of exponents.

When you multiply two expressions with the same base, you add the exponents:

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

Now we need to add the fractions \( \frac{3}{5} \) and \( \frac{1}{6} \). To do this, we need a common denominator. The least common multiple of 5 and 6 is 30. We can convert both fractions as follows:

\[ \frac{3}{5} = \frac{3 \cdot 6}{5 \cdot 6} = \frac{18}{30} \] \[ \frac{1}{6} = \frac{1 \cdot 5}{6 \cdot 5} = \frac{5}{30} \]

Now we can add the two fractions:

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

Thus, we can express the left-hand side as:

\[ x^{\frac{3}{5}} \cdot x^{\frac{1}{6}} = x^{\frac{23}{30}} \]

Since the left-hand side is equal to \( x^a \), we have:

\[ x^{\frac{23}{30}} = x^a \]

Therefore, we can conclude:

\[ a = \frac{23}{30} \]

Thus, the value of \( a \) is \( \frac{23}{30} \).

Answer 2

so its a 23/30?

Answer 3

Yes, that is correct! The value of \( a \) in the equation \( x^{\frac{3}{5}} \cdot x^{\frac{1}{6}} = x^a \) is indeed \( \frac{23}{30} \).

Answer 4

how about this one x
6
5
​

x
2
=
x

Answer 5

To evaluate the left-hand side of the equation \( x^{\frac{6}{5}} \cdot x^2 \) and express it in the form \( x^a \), we use the property of exponents that states:

\[ x^m \cdot x^n = x^{m+n} \]

In this case, we can rewrite the expression:

\[ x^{\frac{6}{5}} \cdot x^2 = x^{\frac{6}{5} + 2} \]

Next, we need to express \( 2 \) as a fraction to combine the exponents. We can write \( 2 \) as \( \frac{10}{5} \):

\[ x^{\frac{6}{5} + 2} = x^{\frac{6}{5} + \frac{10}{5}} = x^{\frac{6 + 10}{5}} = x^{\frac{16}{5}} \]

Thus, the left-hand side simplifies to:

\[ x^{\frac{16}{5}} \]

Now, we can equate it to \( x \):

\[ x^{\frac{16}{5}} = x^1 \]

To find the value of \( a \), we set the exponents equal to each other, which gives us:

\[ \frac{16}{5} = 1 \]

To find the exponent equal to 1, we can express 1 as a fraction with the same denominator:

\[ 1 = \frac{5}{5} \]

Thus, we have:

\[ \frac{16}{5} = \frac{5}{5} \]

This indicates that both sides cannot be equal for any value of \( x \) unless \( x = 0 \). Since \( x \) must not equal zero in general settings, the left-hand side cannot equal \( x \) based on the left-hand side evaluation.

In the case that you wanted the value of \( a \) where:

\[ x^{\frac{16}{5}} = x^a, \]

it indicates:

\[ a = \frac{16}{5}. \]

So, the exponent \( a \) in your original equation should be \( \frac{16}{5} \).

Answer 6

Evaluate the left hand side to find the value of a,a in the equation in simplest form.
x, start superscript, start fraction, 5, divided by, 6, end fraction, end superscript, x, squared, equals, x, to the power a
x
6
5
​

x
2
=
x
a

Answer 7

To evaluate the left-hand side of the equation \( x^{\frac{5}{6}} \cdot x^2 \) and express it in the form \( x^a \), we will use the property of exponents that states:

\[ x^m \cdot x^n = x^{m+n} \]

Applying this property, we can rewrite the left-hand side:

\[ x^{\frac{5}{6}} \cdot x^2 = x^{\frac{5}{6} + 2} \]

Next, we express \( 2 \) as a fraction to combine the exponents. We can write \( 2 \) as \( \frac{12}{6} \):

\[ 2 = \frac{12}{6} \]

Now we can add the two fractions:

\[ \frac{5}{6} + 2 = \frac{5}{6} + \frac{12}{6} = \frac{5 + 12}{6} = \frac{17}{6} \]

Thus, we can express the left-hand side as:

\[ x^{\frac{5}{6}} \cdot x^2 = x^{\frac{17}{6}} \]

Now, we can equate this to \( x^a \):

\[ x^{\frac{17}{6}} = x^a \]

From this equality, we can conclude:

\[ a = \frac{17}{6} \]

Therefore, the value of \( a \) is \( \frac{17}{6} \).

Answer 8

Express left parenthesis, 32, to the power 4 , right parenthesis, start superscript, one fifth, end superscript(32
4
)
5
1
​

in simplest radical form

Answer 9

To express \( (32^4)^{\frac{1}{5}} \) in simplest radical form, we can start by applying the property of exponents that states:

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

Applying this to our expression:

\[ (32^4)^{\frac{1}{5}} = 32^{4 \cdot \frac{1}{5}} = 32^{\frac{4}{5}} \]

Now, \( 32^{\frac{4}{5}} \) can be rewritten in terms of radicals. The exponent \( \frac{4}{5} \) can be interpreted as follows:

\[ 32^{\frac{4}{5}} = \sqrt[5]{32^4} \]

Next, we can simplify \( 32^4 \). Since \( 32 = 2^5 \), we can rewrite \( 32^4 \) as:

\[ 32^4 = (2^5)^4 = 2^{5 \cdot 4} = 2^{20} \]

Now substituting this back into the expression:

\[ 32^{\frac{4}{5}} = \sqrt[5]{2^{20}} \]

Using the property of exponents again, we can simplify:

\[ \sqrt[5]{2^{20}} = 2^{\frac{20}{5}} = 2^4 \]

Finally, since \( 2^4 = 16 \), we have:

\[ (32^4)^{\frac{1}{5}} = 16 \]

Thus, the simplest radical form of \( (32^4)^{\frac{1}{5}} \) is:

\[ 16 \]

Answer 10

Express left parenthesis, 6, start superscript, one quarter, end superscript, right parenthesis, cubed(6
4
1
​

)
3
in simplest radical form

Answer 11

To express \( \left(6^{\frac{1}{4}}\right)^3 \) in simplest radical form, we can use the property of exponents that states:

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

Applying this property to our expression:

\[ \left(6^{\frac{1}{4}}\right)^3 = 6^{\frac{1}{4} \cdot 3} = 6^{\frac{3}{4}} \]

Next, we can express \( 6^{\frac{3}{4}} \) in terms of radicals. The exponent \( \frac{3}{4} \) can be interpreted as:

\[ 6^{\frac{3}{4}} = \sqrt[4]{6^3} \]

Now, we need to evaluate \( 6^3 \):

\[ 6^3 = 216 \]

So we have:

\[ 6^{\frac{3}{4}} = \sqrt[4]{216} \]

Now, let's simplify \( \sqrt[4]{216} \). First, we can factor \( 216 \) into its prime factors:

\[ 216 = 6^3 = 2^3 \cdot 3^3 = 2^3 \cdot 3^3 \]

Now, we write \( 216 \) in a way that's easier to work with in terms of the fourth root:

\[ 216 = (2 \cdot 3)^3 = 6^3 \]

Now, the fourth root of \( 216 \):

\[ \sqrt[4]{216} \text{ does not simplify neatly, but we can write it as } \sqrt[4]{2^3 \cdot 3^3}. \]

For simplicity:

\[ \sqrt[4]{2^3 \cdot 3^3} = \sqrt[4]{(2 \cdot 3)^3 \cdot (2 \cdot 3)} = \sqrt[4]{6^3} \cdot \sqrt[4]{6} = \sqrt[4]{6^4 \cdot 6^{-1}} = 6^{3/4} \cdot 6^{-1/4} \]

Thus:

\[ 6^{\frac{3}{4}} = \sqrt[4]{216} \text{ or } 6^{3/4}. \]

For simplicity, we can summarize:

\[ \sqrt[4]{216} \]

Since \( \sqrt[4]{216} \) does not simplify to a whole number, we can summarize the final answer:

\[ \sqrt[4]{216} \text{ or } 6^{\frac{3}{4}}. \]

In simplest radical form, the answer is:

\[ \sqrt[4]{216} \] or \( 6^{\frac{3}{4}} \).