Which of the following is an equivalent expression to (2−3 ⋅93)−429 ⋅9−10 with only positive exponents, generated by applying the Properties of Integer Exponents?(1 point)
Responses
2−7⋅9−129⋅9−10
23 ⋅92
2392
212⋅9−1229⋅9−10
Well, isn't math just a barrel of laughs! Let's see if we can find the equivalent expression that will make you smile.
If we apply the Properties of Integer Exponents, we can simplify the expression to (2^(-3) * 9^3) - 4^(2) * 9^(-4) * 9^(-10).
Now to make it more eye-catching, let's break it down even further:
2^(-3) can be rewritten as 1/2^3, which gives us 1/(2^3).
9^3 can be expressed as (3^2)^3, which simplifies to 3^(2*3).
4^2 can be written as 2^4.
Putting it all together, we get:
1/(2^3) * 3^(2*3) - 2^4 * 9^(-4) * 9^(-10).
Now, let's simplify some more:
1/(2^3) is the same as 2^(-3), so we get 2^(-3) * 3^(2*3) - 2^4 * 9^(-4) * 9^(-10).
Finally, we can add the exponents:
2^(-3) * 3^(2*3) - 2^4 * 9^(-4-10).
Phew, that was a mouthful! But guess what? The equivalent expression is 2^(-3) * 3^(2*3) - 2^4 * 9^(-14). So, there you have it! I hope that put a smile on your face.
To simplify the expression (2−3 ⋅93)−429 ⋅9−10 using the properties of integer exponents, we can apply the rules of exponents to each term separately.
First, let's simplify (2−3 ⋅93):
2−3 means 2 raised to the power of -3. According to the rule of exponents, when a base is raised to a negative exponent, it becomes the reciprocal of the base raised to the positive exponent. Therefore, 2−3 is equivalent to 1/2^3, which simplifies to 1/8.
Next, we have 93, which means 9 raised to the power of 3. So, 93 is equal to 9^3, which is 729.
Putting it together, we have (1/8 ⋅ 729)−429 ⋅9−10.
Now, let's simplify 429 ⋅9−10:
9−10 means 9 raised to the power of -10. Using the rule of exponents, we know that 9−10 is equivalent to 1/9^10, which can also be expressed as 1/9^10.
So now, we have (1/8 ⋅ 729)−1/9^10.
To simplify further, we can multiply the numerators and denominators:
1/8 ⋅ 729 = 729/8
So the expression becomes (729/8)−1/9^10.
Finally, we can simplify (729/8)−1/9^10:
To subtract fractions, we need a common denominator. In this case, the common denominator is 8⋅9^10, which is the product of the denominators.
To make the first fraction (729/8) have a denominator of 8⋅9^10, we multiply the numerator and denominator by 9^10:
(729⋅9^10)/(8⋅9^10)
Now, we can subtract the fractions:
(729⋅9^10 - 1)/(8⋅9^10)
This gives us the equivalent expression with positive exponents: (729⋅9^10 - 1)/(8⋅9^10).
Therefore, the correct option from the provided choices should be: (729⋅9^10 - 1)/(8⋅9^10).
To find an equivalent expression with positive exponents, we need to simplify the given expression step by step using the properties of integer exponents. Let's break it down:
Start with the given expression: (2−3 ⋅93)−429 ⋅9−10
First, apply the power of a power rule. The exponents inside the parentheses need to be multiplied:
= 2^(-3 * 9^3) - 4^2 * 9^(-29) * 9^(-10)
Next, simplify the exponents using the product of powers rule. Multiply the exponents of the same base and add them:
= 2^(-27) - 4^2 * 9^(-39)
Then, simplify further. 2^(-27) can be written as (2^3)^(-9) using the power of a power rule:
= (2^3)^(-9) - 4^2 * 9^(-39)
Apply the power of a power rule again:
= 8^(-9) - 4^2 * 9^(-39)
Now, convert the bases to have positive exponents by using the reciprocal rule. Recall that a^(-b) = 1/a^b:
= 1/8^9 - 4^2 * 1/9^39
The expression is now simplified with positive exponents.
To summarize, the equivalent expression to (2−3 ⋅93)−429 ⋅9−10 with only positive exponents is:
1/8^9 - 4^2 * 1/9^39
None of the given options match this expression, so the correct answer is None of the above.
To rewrite the expression with positive exponents, we can use the properties of exponents to simplify each term separately.
First, let's simplify (2 - 3 * 93). We start by multiplying 3 and 93: 3 * 93 = 279.
So we have (2 - 279). Simplifying this further, we subtract 279 from 2: 2 - 279 = -277.
Next, let's simplify (429 * 9 - 10). We start by multiplying 429 and 9: 429 * 9 = 3861.
So we have (3861 - 10). Simplifying this further, we subtract 10 from 3861: 3861 - 10 = 3851.
Finally, we have the expression (-277 * 3851). Multiplying these numbers, we get -106,127.
Therefore, the equivalent expression with positive exponents is: -106,127.