Simplify square root of 2x^2* square root of 15x^2
The given problem is asking to simplify the mathematical expression involving the square roots of two terms multiplied together. Each term contains an algebraic expression within the square root. The first term is the square root of 2 times x squared (sqrt(2x^2)), and the second term is the square root of 15 times x squared (sqrt(15x^2)). The task is to combine these two square root terms into one simplified expression using algebraic rules for manipulating square roots and exponents.
$\sqrt{2 x^{2}} \cdot \sqrt{15 x^{2}}$
Answer
Solution:
Step 1:
Arrange the factors within the square roots. $\sqrt{2x^{2}} \cdot \sqrt{15x^{2}}$
Step 2:
Extract the square root of $x^{2}$ from the first radical. $x\sqrt{2} \cdot \sqrt{15x^{2}}$
Step 3:
Arrange the factors within the second square root. $x\sqrt{2} \cdot \sqrt{x^{2} \cdot 15}$
Step 4:
Extract the square root of $x^{2}$ from the second radical. $x\sqrt{2} \cdot (x\sqrt{15})$
Step 5:
Combine the $x$ terms by using the properties of exponents.
Step 5.1:
Rearrange the terms. $x \cdot x\sqrt{2} \cdot \sqrt{15}$
Step 5.2:
Combine the $x$ terms. $x^{2}\sqrt{2} \cdot \sqrt{15}$
Step 6:
Multiply the radicals $x^{2}\sqrt{2}$ and $\sqrt{15}$.
Step 6.1:
Apply the product rule for radicals. $x^{2}\sqrt{2 \cdot 15}$
Step 6.2:
Calculate the product inside the radical. $x^{2}\sqrt{30}$
Knowledge Notes:
To simplify the expression $\sqrt{2x^2} \cdot \sqrt{15x^2}$, we use several properties of square roots and exponents:
Product Property of Square Roots: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, which allows us to combine or separate square roots.
Square Root of a Square: $\sqrt{x^2} = x$, assuming $x$ is non-negative. This is because squaring a number and then taking the square root should return the original non-negative number.
Multiplication of Radicals: When multiplying radicals, we can multiply the numbers inside the radicals together under a single radical sign.
Exponent Rules: When multiplying like bases, we add the exponents: $x^m \cdot x^n = x^{m+n}$.
Using these properties, we can simplify the given expression step by step, extracting square roots of perfect squares and combining radicals until we reach the simplest form.
