Simplify 7x square root of 98x^2-x^2 square root of 162
The given problem is a mathematical expression simplification task. It involves algebraic manipulation and simplification of square roots and other algebraic terms. The initial expression includes two parts - one with a multiple of 'x' accompanied by the square root of a product involving a variable squared (98x^2), and the other part subtracting another term that also has an 'x' squared multiplied by the square root of a number (162). To simplify the expression, one would have to apply properties of square roots and combining like terms.
$7 x \sqrt{98 x^{2}} - x^{2} \sqrt{162}$
Answer
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 $2$. $7x\sqrt{7^2x^2 \cdot 2} - x^2\sqrt{162}$
Step:1.1.4 Rewrite $7^2x^2$ 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:
Square Root Simplification: $\sqrt{a^2} = a$ for any non-negative real number $a$.
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 ($7^2$).
Combining Like Terms: Terms that have the same variable raised to the same power can be combined by adding or subtracting their coefficients.
Distributive Property: This property is used to multiply a single term and two or more terms inside a set of parentheses. For example, $a(b + c) = ab + ac$.
Radical Rules: When a term inside a square root is a perfect square, it can be taken out of the square root. For example, $\sqrt{a^2b} = a\sqrt{b}$ if $a$ is non-negative.
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.
