Math / Algebra

QuestionThe expression $\sqrt[6]{2} \cdot \sqrt{2^{5}}$ is equivalent to
Answer $2^{\frac{12}{5}}$ $2^{\frac{3}{8}}$ $2^{\frac{8}{3}}$ $2^{\frac{5}{12}}$ You have up to 9 questions left to raise your score.

Studdy Solution

STEP 1

What is this asking? We need to simplify the given expression with roots and exponents and write it as a power of 2. Watch out! Don't mix up the root indexes and exponents!
Remember, roots are fractional exponents!

STEP 2

1. Rewrite the roots as fractional exponents
2. Combine the exponents
3. Simplify the exponent

STEP 3

Let's rewrite the roots as fractional exponents!
Remember, the sixth root of 2 is the same as 2 raised to the power of one-sixth.
So, $\sqrt[6]{2}$ can be written as $2^{\frac{1}{6}}$ .
This is because taking the sixth root is the inverse operation of raising to the sixth power.

STEP 4

Similarly, the square root of $2^5$ is the same as $2^5$ raised to the power of one-half.
So, $\sqrt{2^5}$ becomes $(2^5)^{\frac{1}{2}}$ , which simplifies to $2^{\frac{5}{2}}$ using the power of a power rule.
Remember, when you raise a power to a power, you multiply the exponents!

STEP 5

Now, our expression looks like this: $2^{\frac{1}{6}} \cdot 2^{\frac{5}{2}}$ .
Since we're multiplying two terms with the same base, we can add their exponents!
This is a fundamental rule of exponents.

STEP 6

So, we have $2^{\frac{1}{6} + \frac{5}{2}}$ .
To add these fractions, we need a common denominator.
The least common denominator of 6 and 2 is 6.
Let's convert $\frac{5}{2}$ to a fraction with a denominator of 6.
We multiply both the numerator and denominator by 3, giving us $\frac{15}{6}$ .

STEP 7

Now, we can add the exponents: $\frac{1}{6} + \frac{15}{6} = \frac{1+15}{6} = \frac{16}{6}$ .

STEP 8

We have $2^{\frac{16}{6}}$ .
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
So, $\frac{16}{6}$ simplifies to $\frac{16 \div 2}{6 \div 2} = \frac{8}{3}$ .

STEP 9

Therefore, our final simplified expression is $2^{\frac{8}{3}}$ .
Awesome!

STEP 10

The expression $\sqrt[6]{2} \cdot \sqrt{2^{5}}$ simplifies to $2^{\frac{8}{3}}$ .

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QuestionThe expression $\sqrt[6]{2} \cdot \sqrt{2^{5}}$ is equivalent to
Answer $2^{\frac{12}{5}}$ $2^{\frac{3}{8}}$ $2^{\frac{8}{3}}$ $2^{\frac{5}{12}}$ You have up to 9 questions left to raise your score.

Studdy Solution

STEP 1

What is this asking? We need to simplify the given expression with roots and exponents and write it as a power of 2. Watch out! Don't mix up the root indexes and exponents!
Remember, roots are fractional exponents!

STEP 2

1. Rewrite the roots as fractional exponents
2. Combine the exponents
3. Simplify the exponent

STEP 3

Let's rewrite the roots as fractional exponents!
Remember, the sixth root of 2 is the same as 2 raised to the power of one-sixth.
So, $\sqrt[6]{2}$ can be written as $2^{\frac{1}{6}}$ .
This is because taking the sixth root is the inverse operation of raising to the sixth power.

STEP 4

Similarly, the square root of $2^5$ is the same as $2^5$ raised to the power of one-half.
So, $\sqrt{2^5}$ becomes $(2^5)^{\frac{1}{2}}$ , which simplifies to $2^{\frac{5}{2}}$ using the power of a power rule.
Remember, when you raise a power to a power, you multiply the exponents!

STEP 5

Now, our expression looks like this: $2^{\frac{1}{6}} \cdot 2^{\frac{5}{2}}$ .
Since we're multiplying two terms with the same base, we can add their exponents!
This is a fundamental rule of exponents.

STEP 6

So, we have $2^{\frac{1}{6} + \frac{5}{2}}$ .
To add these fractions, we need a common denominator.
The least common denominator of 6 and 2 is 6.
Let's convert $\frac{5}{2}$ to a fraction with a denominator of 6.
We multiply both the numerator and denominator by 3, giving us $\frac{15}{6}$ .

STEP 7

Now, we can add the exponents: $\frac{1}{6} + \frac{15}{6} = \frac{1+15}{6} = \frac{16}{6}$ .

STEP 8

We have $2^{\frac{16}{6}}$ .
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
So, $\frac{16}{6}$ simplifies to $\frac{16 \div 2}{6 \div 2} = \frac{8}{3}$ .

STEP 9

Therefore, our final simplified expression is $2^{\frac{8}{3}}$ .
Awesome!

STEP 10

The expression $\sqrt[6]{2} \cdot \sqrt{2^{5}}$ simplifies to $2^{\frac{8}{3}}$ .

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QuestionThe expression 26⋅25\sqrt[6]{2} \cdot \sqrt{2^{5}}62​⋅25​ is equivalent toAnswer 21252^{\frac{12}{5}}2512​ 2382^{\frac{3}{8}}283​ 2832^{\frac{8}{3}}238​ 25122^{\frac{5}{12}}2125​ You have up to 9 questions left to raise your score.

Studdy Solution

STEP 1

What is this asking? We need to simplify the given expression with roots and exponents and write it as a power of 2. Watch out! Don't mix up the root indexes and exponents!Remember, roots are fractional exponents!

STEP 2

1. Rewrite the roots as fractional exponents2. Combine the exponents3. Simplify the exponent

STEP 3

Let's **rewrite** the roots as fractional exponents!Remember, the sixth root of 2 is the same as 2 raised to the power of one-sixth.So, 26\sqrt[6]{2}62​ can be written as 2162^{\frac{1}{6}}261​.This is because taking the sixth root is the inverse operation of raising to the sixth power.

STEP 4

STEP 5

Now, our expression looks like this: 216⋅2522^{\frac{1}{6}} \cdot 2^{\frac{5}{2}}261​⋅225​.Since we're multiplying two terms with the same base, we can **add** their exponents!This is a fundamental rule of exponents.

STEP 6

STEP 7

Now, we can add the exponents: 16+156=1+156=166\frac{1}{6} + \frac{15}{6} = \frac{1+15}{6} = \frac{16}{6}61​+615​=61+15​=616​.

STEP 8

We have 21662^{\frac{16}{6}}2616​.We can **simplify** this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.So, 166\frac{16}{6}616​ simplifies to 16÷26÷2=83\frac{16 \div 2}{6 \div 2} = \frac{8}{3}6÷216÷2​=38​.

STEP 9

Therefore, our **final** simplified expression is 2832^{\frac{8}{3}}238​.Awesome!

STEP 10

The expression 26⋅25\sqrt[6]{2} \cdot \sqrt{2^{5}}62​⋅25​ simplifies to 2832^{\frac{8}{3}}238​.

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Studdy solves anything!

Start learning now

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QuestionThe expression $\sqrt[6]{2} \cdot \sqrt{2^{5}}$ is equivalent to
Answer $2^{\frac{12}{5}}$ $2^{\frac{3}{8}}$ $2^{\frac{8}{3}}$ $2^{\frac{5}{12}}$ You have up to 9 questions left to raise your score.

What is this asking? We need to simplify the given expression with roots and exponents and write it as a power of 2. Watch out! Don't mix up the root indexes and exponents!
Remember, roots are fractional exponents!

1. Rewrite the roots as fractional exponents
2. Combine the exponents
3. Simplify the exponent

Let's rewrite the roots as fractional exponents!
Remember, the sixth root of 2 is the same as 2 raised to the power of one-sixth.
So, $\sqrt[6]{2}$ can be written as $2^{\frac{1}{6}}$ .
This is because taking the sixth root is the inverse operation of raising to the sixth power.

Now, our expression looks like this: $2^{\frac{1}{6}} \cdot 2^{\frac{5}{2}}$ .
Since we're multiplying two terms with the same base, we can add their exponents!
This is a fundamental rule of exponents.

Now, we can add the exponents: $\frac{1}{6} + \frac{15}{6} = \frac{1+15}{6} = \frac{16}{6}$ .

We have $2^{\frac{16}{6}}$ .
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
So, $\frac{16}{6}$ simplifies to $\frac{16 \div 2}{6 \div 2} = \frac{8}{3}$ .

Therefore, our final simplified expression is $2^{\frac{8}{3}}$ .
Awesome!

The expression $\sqrt[6]{2} \cdot \sqrt{2^{5}}$ simplifies to $2^{\frac{8}{3}}$ .