To write the infinite decimal expansion as a fraction, we can convert it into a geometric series.
Let x = 0.530530530...
Multiplying both sides of the equation by 1000, we get:
1000x = 530.530530...
Subtracting the original equation from the new equation, we get:
1000x - x = 530.530530... - 0.530530530...
Simplifying, we have:
999x = 530
Dividing both sides of the equation by 999, we get:
x = 530/999
Therefore, the infinite decimal expansion 0.530530530... is equal to the fraction 530/999.
Write the following infinite decimal expansion as a fraction. You do not need to simplify.
0.53‾0.\overline{53}0.
53
11 answers
The hypotenuse of a triangle was 3 cm and the hypotenuse of the final triangle was 6 cm. What is the scale factor?
The scale factor is the ratio of the lengths of corresponding sides in the two similar triangles.
Let's call the length of the original side of the triangle x, and the corresponding side in the final triangle y.
In the original triangle, we have:
x² + (3 cm)² = (hypotenuse of the original triangle)²
x² + 9 = 9
x² = 0
In the final triangle, we have:
y² + (6 cm)² = (hypotenuse of the final triangle)²
y² + 36 = 36
y² = 0
Since the length of both sides is 0, we can say that x = y = 0.
Therefore, the scale factor is 0.
Let's call the length of the original side of the triangle x, and the corresponding side in the final triangle y.
In the original triangle, we have:
x² + (3 cm)² = (hypotenuse of the original triangle)²
x² + 9 = 9
x² = 0
In the final triangle, we have:
y² + (6 cm)² = (hypotenuse of the final triangle)²
y² + 36 = 36
y² = 0
Since the length of both sides is 0, we can say that x = y = 0.
Therefore, the scale factor is 0.
Simplify 5^10 ⋅ 5 and write your answer in exponential form.
To simplify and write the answer in exponential form, we can add the exponents since we are multiplying powers with the same base:
5^10 ⋅ 5 = 5^(10 + 1)
Simplifying the exponent:
5^(10 + 1) = 5^11
Therefore, the simplified form in exponential form is 5^11.
5^10 ⋅ 5 = 5^(10 + 1)
Simplifying the exponent:
5^(10 + 1) = 5^11
Therefore, the simplified form in exponential form is 5^11.
Simplify 7^15 . 7^-5 and write your answer in exponential form
To simplify 7^15 ⋅ 7^-5 and write the answer in exponential form, we can subtract the exponents since we are dividing powers with the same base:
7^15 ⋅ 7^-5 = 7^(15 - 5)
Simplifying the exponent:
7^(15 - 5) = 7^10
Therefore, the simplified form in exponential form is 7^10.
7^15 ⋅ 7^-5 = 7^(15 - 5)
Simplifying the exponent:
7^(15 - 5) = 7^10
Therefore, the simplified form in exponential form is 7^10.
Fully simplify and write your answer in exponential form with positive exponents only.
6^-3 . 6 . 6^-6
6^-3 . 6 . 6^-6
To fully simplify the expression 6^-3 ⋅ 6 ⋅ 6^-6 and write the answer in exponential form with positive exponents only, we can combine the terms with the same base and add the exponents:
6^-3 ⋅ 6 ⋅ 6^-6
Using the rules of exponents, we can rewrite any negative exponent as the reciprocal of the base raised to the positive exponent.
6^-3 can be rewritten as 1/6^3 and 6^-6 can be rewritten as 1/6^6.
Substituting these values, we have:
(1/6^3) ⋅ 6 ⋅ (1/6^6)
Simplifying the expression:
= 6/6^3 ⋅ 1/6^6
= 6/216 ⋅ 1/46656
= 1/36 ⋅ 1/46656
To multiply the fractions, we can multiply the numerators and multiply the denominators:
= (1 ⋅ 1)/(36 ⋅ 46656)
= 1/1679616
Therefore, the fully simplified expression in exponential form with positive exponents only is 1/1679616.
6^-3 ⋅ 6 ⋅ 6^-6
Using the rules of exponents, we can rewrite any negative exponent as the reciprocal of the base raised to the positive exponent.
6^-3 can be rewritten as 1/6^3 and 6^-6 can be rewritten as 1/6^6.
Substituting these values, we have:
(1/6^3) ⋅ 6 ⋅ (1/6^6)
Simplifying the expression:
= 6/6^3 ⋅ 1/6^6
= 6/216 ⋅ 1/46656
= 1/36 ⋅ 1/46656
To multiply the fractions, we can multiply the numerators and multiply the denominators:
= (1 ⋅ 1)/(36 ⋅ 46656)
= 1/1679616
Therefore, the fully simplified expression in exponential form with positive exponents only is 1/1679616.
Simplify the following and leave your answer in exponential form.
7^17/7^-2
7^17/7^-2
To simplify the expression 7^17/7^-2, we can use the rules of exponents to divide the terms with the same base and subtract the exponents:
7^17/7^-2 = 7^(17-(-2))
Simplifying the exponent:
7^(17-(-2)) = 7^(17+2)
= 7^19
Therefore, the simplified form in exponential form is 7^19.
7^17/7^-2 = 7^(17-(-2))
Simplifying the exponent:
7^(17-(-2)) = 7^(17+2)
= 7^19
Therefore, the simplified form in exponential form is 7^19.
1 answer