Example Question - solving cube root
Here are examples of questions we've helped users solve.
Solving Cube Root of 72
To solve the expression given in the image, which is the cube root of 72, you need to find the number that multiplied by itself three times equals 72. The cube root of a number \( x \) is written as \( \sqrt[3]{x} \). \( \sqrt[3]{72} \) can be simplified by looking for the largest cube number factor of 72. Let's break down 72 into its prime factors: \( 72 = 2 \times 2 \times 2 \times 3 \times 3 \) You can group the prime factors into triples: \( 72 = (2 \times 2 \times 2) \times (3 \times 3) \) \( 72 = 2^3 \times 3^2 \) Now we can take the cube root: \( \sqrt[3]{72} = \sqrt[3]{2^3 \times 3^2} \) Since \( \sqrt[3]{2^3} = 2 \) and \( 3^2 \) does not have a perfect cube in this factorization, our final expression becomes: \( \sqrt[3]{72} = 2 \times \sqrt[3]{3^2} \) \( \sqrt[3]{72} = 2 \times \sqrt[3]{9} \) So the cube root of 72 is \( 2 \times \sqrt[3]{9} \). This is the simplest radical form without using decimal or approximate values. If you need a decimal approximation, you would have to use a calculator. The approximate decimal value of \( \sqrt[3]{72} \) is about 4.1602.
Solving Cube Root of 81x^10y^8
The expression in the image is the cube root of \((81x^{10}y^8)\), which can be written as \((81x^{10}y^8)^{\frac{1}{3}}\). To simplify this expression, we take the cube root of each factor separately: 1. The cube root of 81, which is \(3^4\), is 3, because \(3^3 = 27\) and \(3^3 \times 3 = 81\). 2. The cube root of \(x^{10}\) can be simplified by dividing the exponent by 3. This gives us \(x^{\frac{10}{3}}\) or \(x^3 \times x^{\frac{1}{3}}\). 3. The cube root of \(y^8\) can be simplified by dividing the exponent by 3. This gives us \(y^{\frac{8}{3}}\) or \(y^2 \times y^{\frac{2}{3}}\). Putting it all together, you get: \(3x^3x^{\frac{1}{3}}y^2y^{\frac{2}{3}}\) You can also leave it as \(3x^{\frac{10}{3}}y^{\frac{8}{3}}\) if you prefer. Both forms are mathematically equivalent.
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