To find the ratio of the length to the width of the rectangle, we can simply divide the length by the width.
Given that the length of the rectangle is \(x^3 \cdot x^4\) inches, we can combine the exponents because the base is the same (the base is x). When multiplying powers with the same base, we add the exponents:
Length = \(x^3 \cdot x^4 = x^{3+4} = x^7\) inches.
The width of the rectangle is given as \(x^7\) inches.
Now, we can write the ratio of the length to the width as:
\[\frac{\text{length}}{\text{width}} = \frac{x^7}{x^7}.\]
Since the bases are the same and both exponents are equal, dividing powers with the same base allows us to subtract the exponents:
\[\frac{x^7}{x^7} = x^{7-7} = x^0.\]
We know that any non-zero number raised to the power of 0 is equal to 1:
\(x^0 = 1.\)
So, the ratio of the length to the width of the rectangle is 1:1. This means that the rectangle is actually a square because its length and width are the same.
A rectangle has a length of x^3 x^4 inches and a width of x^7 inches. Which expression represents the ratio of the length of the rectangle to the width of the rectangle?
3 answers
A rectangle has a length of x^3y^4 inches and a width of xy^7 inches. Which expression represents the ratio of the length of the rectangle to the width of the rectangle?
To find the ratio of the length to the width of the rectangle, we divide the length by the width.
Given that:
Length (L) = \(x^3y^4\) inches,
Width (W) = \(xy^7\) inches.
The ratio of the length to the width is:
\[\frac{L}{W} = \frac{x^3y^4}{xy^7}.\]
Now, divide the powers with the same base. For the base x, we subtract the exponent of the denominator from the exponent of the numerator:
\(x^3 \div x = x^{3-1} = x^2.\)
For the base y, we subtract the exponent of the denominator from the exponent of the numerator:
\(y^4 \div y^7 = y^{4-7} = y^{-3}.\)
Therefore, the ratio \(L/W\) simplifies to:
\[\frac{L}{W} = x^2y^{-3}.\]
Remember that negative exponents denote the reciprocal of the base raised to the positive exponent. Therefore, \(y^{-3} = 1/y^3.\)
So the ratio can also be presented without negative exponents as:
\[\frac{L}{W} = \frac{x^2}{y^3}.\]
This is the simplified expression that represents the ratio of the length to the width of the rectangle.
Given that:
Length (L) = \(x^3y^4\) inches,
Width (W) = \(xy^7\) inches.
The ratio of the length to the width is:
\[\frac{L}{W} = \frac{x^3y^4}{xy^7}.\]
Now, divide the powers with the same base. For the base x, we subtract the exponent of the denominator from the exponent of the numerator:
\(x^3 \div x = x^{3-1} = x^2.\)
For the base y, we subtract the exponent of the denominator from the exponent of the numerator:
\(y^4 \div y^7 = y^{4-7} = y^{-3}.\)
Therefore, the ratio \(L/W\) simplifies to:
\[\frac{L}{W} = x^2y^{-3}.\]
Remember that negative exponents denote the reciprocal of the base raised to the positive exponent. Therefore, \(y^{-3} = 1/y^3.\)
So the ratio can also be presented without negative exponents as:
\[\frac{L}{W} = \frac{x^2}{y^3}.\]
This is the simplified expression that represents the ratio of the length to the width of the rectangle.
1 answer