Question

using the power rule of exponents to simplify the expression

(x^6)^5

Answers

GPT 3.5
To simplify the expression, we can use the power rule of exponents, which states that (a^m)^n = a^(m*n).

In this case, we have (x^6)^5. Applying the power rule, we multiply the exponents:

(x^6)^5 = x^(6*5) = x^30

Therefore, the expression simplifies to x^30.
which of the following is equivalent to 3^-4? 1 point
1/3^4
1/81
81
-81
GPT 3.5
To find an equivalent expression for 3^-4, we need to understand that a negative exponent indicates a reciprocal. Specifically, for any non-zero number a, a^(-n) = 1/a^n.

In this case, we have 3^-4. Applying the rule mentioned above, we have 1/3^4.

Therefore, the equivalent expression for 3^-4 is 1/81.
are you sure
GPT 3.5
Apologies for the mistake in my previous response. Let's correct it.

To find an equivalent expression for 3^-4, we need to raise 3 to the power of -4.

Using the rule that a^(-n) = 1/a^n, we have:

3^-4 = 1/3^4

Therefore, the correct equivalent expression for 3^-4 is 1/81.
apply the properties of integer exponents to generate an equivalent to (5^3)^6 x 5^17/10^2 with only positive exponents, the solove the expression
GPT 3.5
To generate an equivalent expression with only positive exponents, we can use the property (a^m)^n = a^(m*n) and the property a^(-n) = 1/a^n.

Let's start by applying the property (a^m)^n = a^(m*n) to simplify (5^3)^6:

(5^3)^6 = 5^(3*6) = 5^18

Next, we can rewrite 5^17 as (5^17)/(10^0) to express it with a positive exponent.

Now, let's rewrite 10^2 as (10^2)^1 to make the exponent positive.

Putting it all together, we have:

(5^3)^6 * 5^17/10^2 = 5^18 * (5^17)/(10^0) * (10^2)^1

Now, we can combine the terms with the same base using the property a^m * a^n = a^(m+n):

= 5^(18+17) / (10^0 * 10^2)

= 5^35 / (10^0 * 10^2)

Since any number to the power of 0 is equal to 1, we have:

= 5^35 / (1 * 10^2)

= 5^35 / 10^2

Now, we can simplify the expression further:

= (5^35 / 10^2) = (5^35 / 100)

Finally, since 100 is a perfect square, we can further simplify by dividing both the numerator and denominator by 100:

= 5^35 / 100 = 5^35 / (10^2) = (5^35 / 10^2) = (5^35 / 10^2) = 5^35/1 = 5^35.

Therefore, the simplified expression is 5^35.
are you sure?
GPT 3.5
I apologize for the mistake in my previous response. Let's correct it.

To generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents:

First, let's simplify (5^3)^6 using the property (a^m)^n = a^(m*n):

(5^3)^6 = 5^(3*6) = 5^18

Next, let's rewrite 5^17 as (5^17)/(10^0) and rewrite 10^2 as (10^2)^1 to make the exponents positive:

(5^18)*(5^17)/(10^0)*(10^2)^1

Now, let's use the property a^m * a^n = a^(m+n) to combine the terms with the same base:

= 5^(18+17)/(10^0 * 10^2)

= 5^(35)/(10^0 * 10^2)

Since any number to the power of 0 is equal to 1, we have:

= 5^(35)/(1 * 10^2)

= 5^(35)/(10^2)

Now, we can simplify further by dividing both the numerator and denominator by 10^2:

= (5^(35-2))/(1)

= 5^(35-2)

= 5^33

Therefore, the simplified expression is 5^33. Apologies for the error in my previous response.

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