Math / Algebra

QuestionUse properties of exponents to simply the expression. Write all answers with positive exponents only. (Assume all variables are nonzero.) $\left(\frac{a^{9}}{a^{3}}\right)^{3}=$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to simplify an expression with exponents and make sure the final answer only has positive exponents. Watch out! It's easy to mess up exponent rules, so let's be extra careful and go step by step!

STEP 2

1. Simplify inside the parentheses.
2. Apply the outer exponent.

STEP 3

Alright, let's look inside those parentheses!
We've got $\frac{a^{9}}{a^{3}}$ .
When we divide terms with the same base, we subtract the exponents.
It's like saying "I have nine a's on top and three a's on the bottom, so if I pair them up, how many a's are left on top?"

STEP 4

So, we have $a^{9-3} = a^{6}$ .
See? Six a's left on top!
So, our expression now looks like $(a^{6})^{3}$ .

STEP 5

Now, we have an exponent raised to another exponent – it's power to the power time!
In this case, we multiply the exponents.
Think of it like this: $(a^{6})^{3}$ means we have three groups of six a's multiplied together.

STEP 6

So, we multiply the exponents: $a^{6 \cdot 3} = a^{18}$ .
Boom!

STEP 7

Our simplified expression is $a^{18}$ .
And look at that, only positive exponents!
We crushed it!

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QuestionUse properties of exponents to simply the expression. Write all answers with positive exponents only. (Assume all variables are nonzero.) $\left(\frac{a^{9}}{a^{3}}\right)^{3}=$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to simplify an expression with exponents and make sure the final answer only has positive exponents. Watch out! It's easy to mess up exponent rules, so let's be extra careful and go step by step!

STEP 2

1. Simplify inside the parentheses.
2. Apply the outer exponent.

STEP 3

Alright, let's look inside those parentheses!
We've got $\frac{a^{9}}{a^{3}}$ .
When we divide terms with the same base, we subtract the exponents.
It's like saying "I have nine a's on top and three a's on the bottom, so if I pair them up, how many a's are left on top?"

STEP 4

So, we have $a^{9-3} = a^{6}$ .
See? Six a's left on top!
So, our expression now looks like $(a^{6})^{3}$ .

STEP 5

Now, we have an exponent raised to another exponent – it's power to the power time!
In this case, we multiply the exponents.
Think of it like this: $(a^{6})^{3}$ means we have three groups of six a's multiplied together.

STEP 6

So, we multiply the exponents: $a^{6 \cdot 3} = a^{18}$ .
Boom!

STEP 7

Our simplified expression is $a^{18}$ .
And look at that, only positive exponents!
We crushed it!

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QuestionUse properties of exponents to simply the expression. Write all answers with positive exponents only. (Assume all variables are nonzero.) (a9a3)3=\left(\frac{a^{9}}{a^{3}}\right)^{3}=(a3a9​)3= □\square□

Studdy Solution

STEP 1

What is this asking? We need to simplify an expression with exponents and make sure the final answer only has positive exponents. Watch out! It's easy to mess up exponent rules, so let's be extra careful and go step by step!

STEP 2

1. Simplify inside the parentheses.2. Apply the outer exponent.

STEP 3

Alright, let's look inside those parentheses!We've got a9a3\frac{a^{9}}{a^{3}}a3a9​.When we divide terms with the same base, we subtract the exponents.It's like saying "I have nine *a*'s on top and three *a*'s on the bottom, so if I pair them up, how many *a*'s are left on top?"

STEP 4

So, we have a9−3=a6a^{9-3} = a^{6}a9−3=a6.See? Six *a*'s left on top!So, our expression now looks like (a6)3(a^{6})^{3}(a6)3.

STEP 5

Now, we have an exponent raised to another exponent – it's power to the power time!In this case, we multiply the exponents.Think of it like this: (a6)3(a^{6})^{3}(a6)3 means we have three groups of six *a*'s multiplied together.

STEP 6

So, we multiply the exponents: a6⋅3=a18a^{6 \cdot 3} = a^{18}a6⋅3=a18.Boom!

STEP 7

Our simplified expression is a18a^{18}a18.And look at that, only positive exponents!We crushed it!

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionUse properties of exponents to simply the expression. Write all answers with positive exponents only. (Assume all variables are nonzero.) $\left(\frac{a^{9}}{a^{3}}\right)^{3}=$ $\square$

1. Simplify inside the parentheses.
2. Apply the outer exponent.

Alright, let's look inside those parentheses!
We've got $\frac{a^{9}}{a^{3}}$ .
When we divide terms with the same base, we subtract the exponents.
It's like saying "I have nine a's on top and three a's on the bottom, so if I pair them up, how many a's are left on top?"

So, we have $a^{9-3} = a^{6}$ .
See? Six a's left on top!
So, our expression now looks like $(a^{6})^{3}$ .

Now, we have an exponent raised to another exponent – it's power to the power time!
In this case, we multiply the exponents.
Think of it like this: $(a^{6})^{3}$ means we have three groups of six a's multiplied together.

So, we multiply the exponents: $a^{6 \cdot 3} = a^{18}$ .
Boom!

Our simplified expression is $a^{18}$ .
And look at that, only positive exponents!
We crushed it!