Simplify square root of 2a* square root of 14a^3* square root of 5a

Solution:

Step:1

Apply the product rule for radicals to combine them: $\sqrt{2a}\cdot \sqrt{14a^3}\cdot \sqrt{5a}$.

Step:2

Calculate the product of $14$ and $5$: $\sqrt{2a\cdot \sqrt{70a^4}}$.

Step:3

Combine like terms by adding exponents of $a$.

Step:3.1

Rearrange to place $a^3$ next to $a$: $\sqrt{2a\cdot \sqrt{70a^{3}a}}$.

Step:3.2

Combine $a^3$ and $a$.

Step:3.2.1

Express $a$ as $a^1$: $\sqrt{2a\cdot \sqrt{70a^3{a}^1}}$.

Step:3.2.2

Apply the exponent rule $a^m\cdot a^n=a^{m+n}$: $\sqrt{2a\cdot \sqrt{70a^4}}$.

Step:3.3

Sum the exponents $3$ and $1$: $\sqrt{2a\cdot \sqrt{70a^4}}$.

Step:4

Express $70a^4$ as $(a^2{)}^2\cdot 70$.

Step:4.1

Rewrite $a^4$ as $(a^2{)}^2$: $\sqrt{2a\cdot \sqrt{70(a^2{)}^2}}$.

Step:4.2

Switch the order of $70$ and $(a^2{)}^2$: $\sqrt{2a\cdot \sqrt{(a^2{)}^2\cdot 70}}$.

Step:5

Extract terms from under the radical sign: $\sqrt{2a\cdot a^2\cdot \sqrt{70}}$.

Step:6

Multiply $a$ by $a^2$ by adding their exponents.

Step:6.1

Place $a^2$ next to $a$: $\sqrt{2a^{2}a\cdot \sqrt{70}}$.

Step:6.2

Multiply $a^2$ by $a$.

Step:6.2.1

Express $a$ as $a^1$: $\sqrt{2a^2{a}^1\cdot \sqrt{70}}$.

Step:6.2.2

Combine exponents using the rule $a^m\cdot a^n=a^{m+n}$: $\sqrt{2a^3\cdot \sqrt{70}}$.

Step:6.3

Add the exponents $2$ and $1$: $\sqrt{2a^3\cdot \sqrt{70}}$.

Step:7

Rearrange $2a^3\cdot \sqrt{70}$ as $a^2\cdot (2a\cdot \sqrt{70})$.

Step:7.1

Factor out $a^2$: $\sqrt{a^2\cdot 2a\cdot \sqrt{70}}$.

Step:7.2

Switch the positions of $2$ and $a^2$: $\sqrt{a^2\cdot 2a\cdot \sqrt{70}}$.

Step:7.3

Enclose $2a\cdot \sqrt{70}$ in parentheses: $\sqrt{a^2\cdot (2a\cdot \sqrt{70})}$.

Step:7.4

Enclose $2a\cdot \sqrt{70}$ in parentheses for clarity: $\sqrt{a^2\cdot (2a\cdot \sqrt{70})}$.

Step:8

Extract terms from under the radical, simplifying to: $a\cdot \sqrt{2a\cdot \sqrt{70}}$.

Knowledge Notes:

The problem involves simplifying a product of radical expressions. The key knowledge points and rules used in the solution are:

1. Product Rule for Radicals: $\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}$. This rule allows us to combine or separate radicals that are being multiplied.

2. Exponent Rules:

   • Power Rule: $(a^m{)}^n=a^{mn}$.

   • Product of Powers Rule: $a^m\cdot a^n=a^{m+n}$.

   These rules help us to manipulate expressions with exponents, particularly when combining like bases.

3. Simplifying Radicals: When a term inside a radical can be expressed as a perfect square (or a higher power that matches the index of the radical), it can be taken out of the radical. For example, $\sqrt{a^2}=a$.

4. Rearranging Terms: Sometimes, it's necessary to rearrange terms to make it easier to apply the product rule for radicals or the exponent rules.

5. Combining Like Terms: When terms have the same base and are multiplied together, we can add their exponents to combine them into a single term.

By applying these rules and knowledge points step by step, the original expression is simplified to its most reduced form.