Simplify the Radical Expression fourth root of 8a^3b* fourth root of 10a^2b^7

Solution:

Step:1

Express $10a^2{b}^7$ as $b^4\cdot (10a^2{b}^3)$.

Step:1.1

Extract $b^4$ from the expression: $\sqrt[4]{8a^{3}b}\cdot \sqrt[4]{10a^2(b^4{b}^3)}$

Step:1.2

Rearrange to place $a^2$ at the front: $\sqrt[4]{8a^{3}b}\cdot \sqrt[4]{10b^4{a}^2{b}^3}$

Step:1.3

Switch the positions of $10$ and $b^4$: $\sqrt[4]{8a^{3}b}\cdot \sqrt[4]{b^4\cdot 10a^2{b}^3}$

Step:1.4

Enclose in brackets: $\sqrt[4]{8a^{3}b}\cdot \sqrt[4]{b^4\cdot 10(a^2{b}^3)}$

Step:1.5

Enclose in brackets again for clarity: $\sqrt[4]{8a^{3}b}\cdot \sqrt[4]{b^4\cdot (10a^2{b}^3)}$

Step:2

Extract terms from under the radical sign: $\sqrt[4]{8a^{3}b}\cdot (|b|\sqrt[4]{10a^2{b}^3})$

Step:3

Combine the two radicals: $\sqrt[4]{8a^{3}b}(|b|\sqrt[4]{10a^2{b}^3})$.

Step:3.1

Apply the product rule for radicals: $\sqrt[4]{10a^2{b}^3(8a^{3}b)}|b|$

Step:3.2

Multiply $8$ by $10$: $\sqrt[4]{80a^2{b}^3(a^{3}b)}|b|$

Step:3.3

Combine the exponents using the power rule $a^m{a}^n=a^{m+n}$: $\sqrt[4]{80a^5{b}^{3}b}|b|$

Step:3.4

Sum the exponents $3$ and $2$: $\sqrt[4]{80a^5{b}^{3}b}|b|$

Step:3.5

Raise $b$ to the power of $1$: $\sqrt[4]{80a^5(b^1{b}^3)}|b|$

Step:3.6

Combine the exponents: $\sqrt[4]{80a^5{b}^4}|b|$

Step:3.7

Sum the exponents $1$ and $3$: $\sqrt[4]{80a^5{b}^4}|b|$

Step:4

Rewrite $80a^5{b}^4$ as $(2ab{)}^4\cdot (5a)$.

Step:4.1

Factor $16$ from $80$: $\sqrt[4]{16(5)a^5{b}^4}|b|$

Step:4.2

Represent $16$ as $2^4$: $\sqrt[4]{2^4\cdot 5a^5{b}^4}|b|$

Step:4.3

Factor out $a^4$: $\sqrt[4]{2^4\cdot 5(a^{4}a)b^4}|b|$

Step:4.4

Rearrange $a$: $\sqrt[4]{2^4\cdot 5(a^4)b^{4}a}|b|$

Step:4.5

Rearrange $5$: $\sqrt[4]{(2^4(a^4))b^4\cdot 5a}|b|$

Step:4.6

Express $(2^4(a^4))b^4$ as $(2ab{)}^4$: $\sqrt[4]{(2ab{)}^4\cdot 5a}|b|$

Step:4.7

Enclose in brackets: $\sqrt[4]{((2ab{)}^4\cdot (5a))}|b|$

Step:5

Extract terms from under the radical: $|2ab|\sqrt[4]{5a}|b|$

Step:6

Eliminate non-negative terms from the absolute value: $2|ab|\sqrt[4]{5a}|b|$

Step:7

Multiply the terms: $2|ab|\sqrt[4]{5a}|b|$.

Step:7.1

To multiply absolute values, multiply the terms inside: $2|b(ab)|\sqrt[4]{5a}$

Step:7.2

Raise $b$ to the power of $1$: $2|a(b^{1}b)|\sqrt[4]{5a}$

Step:7.3

Raise $b$ to the power of $1$ again: $2|a(b^1{b}^1)|\sqrt[4]{5a}$

Step:7.4

Combine the exponents: $2|a{b}^2|\sqrt[4]{5a}$

Step:7.5

Add the exponents $1$ and $1$: $2|a{b}^2|\sqrt[4]{5a}$

Step:8

Eliminate non-negative terms from the absolute value: $2b^2|a|\sqrt[4]{5a}$

Knowledge Notes:

1. Radical Expressions: Expressions that contain roots, such as square roots or fourth roots.

2. Product Rule for Radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$, which allows us to multiply two radicals with the same index.

3. Power Rule for Exponents: $a^m\cdot a^n=a^{m+n}$, which is used to combine like bases with exponents.

4. Absolute Value: The absolute value of a number is its distance from zero on the number line, denoted by $|a|$. For non-negative terms, the absolute value is the term itself.

5. Simplifying Radical Expressions: The process involves factoring out perfect powers, combining like terms under the radical, and then simplifying the expression inside and outside the radical.

6. Factoring: The process of breaking down an expression into its constituent factors, which can be useful when simplifying expressions, especially under a radical.

7. Exponent Laws: These laws are used to manipulate expressions with exponents during the simplification process.

8. Rewriting Expressions: Often, expressions can be rewritten in a form that makes it easier to simplify or factor, such as expressing a number as a power of another number.