Example Question - laws of exponents
Here are examples of questions we've helped users solve.
Solving a Simplification Expression with Variables and Coefficients
To solve the expression \( \frac{10x^8}{5x^4} \), you can simplify it by dividing both the coefficients (the numerical parts) and the variables separately. First, divide the coefficients: \( \frac{10}{5} = 2 \) Next, apply the laws of exponents for dividing like bases: \( x^8 / x^4 = x^{8-4} = x^4 \) Combine these two results to get the final answer: \( 2x^4 \)
Simplified Expression by Dividing Numerical Coefficients and Variables with Exponents
To simplify the given expression, divide the terms in the numerator by the terms in the denominator. The expression is: \[ \frac{7x^3y^{-5}}{21y^6} \] Divide the numerical coefficients: \[ \frac{7}{21} = \frac{1}{3} \] For the variables, use the laws of exponents to divide the terms with the same base: \[ x^{a} \div x^{b} = x^{a-b} \] \[ y^{a} \div y^{b} = y^{a-b} \] Applying the laws of exponents, you get: \[ x^3 \div x^0 = x^{3-0} = x^3 \] (Any term to the power of 0 is 1, which is why the x term in the denominator is considered \(x^0\)) \[ y^{-5} \div y^6 = y^{-5-6} = y^{-11} \] Now put it all together: \[ \frac{1}{3} x^3 y^{-11} \] The negative exponent indicates the reciprocal: \[ y^{-11} = \frac{1}{y^{11}} \] Thus, the simplified expression is: \[ \frac{x^3}{3y^{11}} \]
Solving Algebraic Expression with Exponents
The image is quite blurry, but it appears to show an algebraic expression involving exponents, specifically: \[ m^7 \div m^3 \] To solve this, you can use the laws of exponents which state that when you divide like bases you subtract the exponents. Therefore: \[ m^7 \div m^3 = m^{7 - 3} = m^4 \]
Simplifying Algebraic Expressions with Exponents
To simplify the given expression, you'll need to use the laws of exponents. Let's simplify the numerator and the denominator separately first, and then we'll see if we can simplify further. The expression given is: \( \frac{10x^{5n+4} + 4.125x^{5n+2}}{3x^{5n+3} - 20x^{5n+1}} \) Numerator: \( 10x^{5n+4} + 4.125x^{5n+2} \) can be factored by taking out the common factor of \( x^{5n+2} \): \( = x^{5n+2} (10x^2 + 4.125) \) Denominator: \( 3x^{5n+3} - 20x^{5n+1} \) similarly can be factored by taking out the common factor of \( x^{5n+1} \): \( = x^{5n+1} (3x^2 - 20) \) Now the expression becomes: \( \frac{x^{5n+2} (10x^2 + 4.125)}{x^{5n+1} (3x^2 - 20)} \) Next, we'll divide the exponents: Since \( x^{5n+2} \) is divided by \( x^{5n+1} \), we subtract the exponents of like bases: \( x^{5n+2} ÷ x^{5n+1} = x^{(5n+2) - (5n+1)} = x^{1} = x \) Now we have: \( \frac{x(10x^2 + 4.125)}{3x^2 - 20} \) The expression is simplified to: \( \frac{10x^3 + 4.125x}{3x^2 - 20} \) This is as simple as the expression can get without further information about \( x \) or \( n \). If certain values are given for these variables, then numerical simplification could proceed. Otherwise, this is the simplified algebraic expression.
Simplifying Algebraic Fraction with Exponents
To simplify the expression, we start by simplifying both the numerator and the denominator separately. We can simplify by combining like terms and applying the laws of exponents. Given expression: \[ \frac{10x^{n+4} + 125x^{n+2}}{3x^{n+3} - 20x^{n+1}} \] First, let's simplify the numerator and the denominator by factoring out the common x term: Numerator: \[ 10x^{n+4} + 125x^{n+2} = x^{n+2}(10x^2 + 125) \] Denominator: \[ 3x^{n+3} - 20x^{n+1} = x^{n+1}(3x^2 - 20) \] Now, rewrite the expression using the factored terms: \[ \frac{x^{n+2}(10x^2 + 125)}{x^{n+1}(3x^2 - 20)} \] Next, cancel out the common x term from both the numerator and the denominator: since we have \(x^{n+2}\) in the numerator and \(x^{n+1}\) in the denominator, we can divide each by \(x^{n+1}\) so that: \[ x^{n+2} / x^{n+1} = x \] The simplified expression will be: \[ \frac{x(10x^2 + 125)}{(3x^2 - 20)} \] Now, we can't simplify further without knowing the specific value of \(n\), as the remaining terms in the parentheses do not have a common x term with which to factor out or cancel. The simplified expression is: \[ x \cdot \frac{10x^2 + 125}{3x^2 - 20} \]
Simplifying Expression with Exponents
To simplify the expression given in the image, we need to follow the laws of exponents for division and addition. The expression is: \[ \frac{10x^{n+4} + 4.125x^{n+2}}{3x^{n+3} - 20x^{n+1}} \] First, factor out the common terms involving \(x\) from the numerator and the denominator. 1. In the numerator, factor out the smallest power of \(x\) which is \(x^{n+2}\): \[ x^{n+2}(10x^{2} + 4.125) \] 2. In the denominator, factor out the smallest power of \(x\) which is \(x^{n+1}\): \[ x^{n+1}(3x^{2} - 20) \] Now we rewrite the expression: \[ \frac{x^{n+2}(10x^{2} + 4.125)}{x^{n+1}(3x^{2} - 20)} \] Next, cancel out the common \(x^{n+1}\) term: \[ \frac{x(10x^{2} + 4.125)}{3x^{2} - 20} \] Since \(x\) does not cancel anything further in the denominator, we leave the expression as is. The final simplified expression is: \[ \frac{x(10x^{2} + 4.125)}{3x^{2} - 20} \] Depending on the context or further instructions, there might be additional simplification possible, like factoring the quadratic expressions if they are factorable. However, without additional context, this is the simplified form based on the given expression.
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