Simplify -7 square root of 128(1)x^11(1)y^14

Solution:

Step 1:

Multiply $128$ by $1$ to maintain the expression. $-7\sqrt{128\cdot 1\cdot x^{11}\cdot 1\cdot y^{14}}$

Step 2:

Simplify the multiplication of $128$ by $1$. $-7\sqrt{128\cdot x^{11}\cdot y^{14}}$

Step 3:

Express $128\cdot x^{11}\cdot y^{14}$ as $(8\cdot x^5\cdot y^7{)}^2\cdot 2\cdot x$.

Step 3.1:

Extract $64$ from $128$. $-7\sqrt{64\cdot 2\cdot x^{11}\cdot y^{14}}$

Step 3.2:

Represent $64$ as $8^2$. $-7\sqrt{8^2\cdot 2\cdot x^{11}\cdot y^{14}}$

Step 3.3:

Take out $x^{10}$ from $x^{11}$. $-7\sqrt{8^2\cdot 2\cdot (x^{10}\cdot x)\cdot y^{14}}$

Step 3.4:

Rewrite $x^{10}$ as $(x^5{)}^2$. $-7\sqrt{8^2\cdot 2\cdot ((x^5{)}^2\cdot x)\cdot y^{14}}$

Step 3.5:

Rewrite $y^{14}$ as $(y^7{)}^2$. $-7\sqrt{8^2\cdot 2\cdot ((x^5{)}^2\cdot x)\cdot (y^7{)}^2}$

Step 3.6:

Rearrange $x$. $-7\sqrt{8^2\cdot 2\cdot (x^5{)}^2\cdot (y^7{)}^2\cdot x}$

Step 3.7:

Rearrange $2$. $-7\sqrt{8^2\cdot ((x^5{)}^2)\cdot (y^7{)}^2\cdot 2\cdot x}$

Step 3.8:

Combine terms under the radical. $-7\sqrt{(8\cdot x^5\cdot y^7{)}^2\cdot 2\cdot x}$

Step 3.9:

Add parentheses to clarify the expression. $-7\sqrt{((8\cdot x^5\cdot y^7{)}^2)\cdot (2\cdot x)}$

Step 4:

Extract terms from under the radical, recognizing that the square root of a square is the value itself. $-7(8\cdot x^5\cdot y^7\cdot \sqrt{2\cdot x})$

Step 5:

Multiply $8$ by $-7$. $-56\cdot x^5\cdot y^7\cdot \sqrt{2\cdot x}$

Knowledge Notes:

To simplify the given expression, we follow a series of algebraic steps:

1. Multiplication of Real Numbers: Multiplying real numbers is straightforward; for example, $128\cdot 1=128$.

2. Square Roots and Exponents: The square root of a number $a^2$ is $a$. Similarly, $\sqrt{x^{2n}}=x^n$ for any integer $n$.

3. Factoring Perfect Squares: Numbers like $64$ are perfect squares because $64=8^2$. This property is used to simplify square roots.

4. Combining Like Terms: Algebraic expressions with the same variables and exponents can be combined, such as $x^{10}\cdot x=x^{11}$.

5. Simplifying Radical Expressions: When a term inside a radical is a perfect square, it can be taken out of the radical. For example, $\sqrt{a^2\cdot b}=a\cdot \sqrt{b}$.

6. Distributive Property: This property is used when multiplying numbers outside the radical with those inside, such as $-7\cdot 8=-56$.

By applying these principles, we can simplify the original expression to its simplest form.