Simplify 7x square root of 98x^2-x^2 square root of 162

Solution:

Simplification Process

Step:1 Break down each mathematical expression.

Step:1.1 Express $98x^2$ as $(7x{)}^2\cdot 2$.

Step:1.1.1 Extract $49$ from $98$. $7x\sqrt{49(2x^2)}-x^2\sqrt{162}$

Step:1.1.2 Represent $49$ as $7^2$. $7x\sqrt{7^2\cdot 2x^2}-x^2\sqrt{162}$

Step:1.1.3 Rearrange $2$. $7x\sqrt{7^2{x}^2\cdot 2}-x^2\sqrt{162}$

Step:1.1.4 Rewrite $7^2{x}^2$ as $(7x{)}^2$. $7x\sqrt{(7x{)}^2\cdot 2}-x^2\sqrt{162}$

Step:1.2 Extract terms from under the square root. $7x(7x\sqrt{2})-x^2\sqrt{162}$

Step:1.3 Combine $x$ with $x$ by adding their powers.

Step:1.3.1 Rearrange $x$. $7(x\cdot x)(7\sqrt{2})-x^2\sqrt{162}$

Step:1.3.2 Multiply $x$ with itself. $7x^2(7\sqrt{2})-x^2\sqrt{162}$

Step:1.4 Multiply $7$ with $7$. $49x^2\sqrt{2}-x^2\sqrt{162}$

Step:1.5 Express $162$ as $9^2\cdot 2$.

Step:1.5.1 Isolate $81$ from $162$. $49x^2\sqrt{2}-x^2\sqrt{81(2)}$

Step:1.5.2 Represent $81$ as $9^2$. $49x^2\sqrt{2}-x^2\sqrt{9^2\cdot 2}$

Step:1.6 Extract terms from under the square root. $49x^2\sqrt{2}-x^2(9\sqrt{2})$

Step:1.7 Multiply $9$ by $-1$. $49x^2\sqrt{2}-9x^2\sqrt{2}$

Step:2 Subtract $9x^2\sqrt{2}$ from $49x^2\sqrt{2}$. $40x^2\sqrt{2}$

Knowledge Notes:

To simplify the given expression, we need to apply several algebraic rules and properties:

1. Square Root Simplification: $\sqrt{a^2}=a$ for any non-negative real number $a$.

2. Factoring: This involves expressing a number as a product of its factors. For example, $98$ can be factored into $49\cdot 2$, where $49$ is a perfect square ($7^2$).

3. Combining Like Terms: Terms that have the same variable raised to the same power can be combined by adding or subtracting their coefficients.

4. Distributive Property: This property is used to multiply a single term and two or more terms inside a set of parentheses. For example, $a(b+c)=ab+ac$.

5. Radical Rules: When a term inside a square root is a perfect square, it can be taken out of the square root. For example, $\sqrt{a^{2}b}=a\sqrt{b}$ if $a$ is non-negative.

6. Exponent Rules: When multiplying powers with the same base, you add the exponents. For example, $x^m\cdot x^n=x^{m+n}$.

Using these rules, we can simplify the given expression step by step, ensuring that we perform operations correctly and simplify square roots by extracting perfect squares.