Simplify 2( square root of 7)^10

Solution:

Step 1: Express the square root as a fractional exponent

Convert ${\left(\sqrt{7}\right)}^{10}$ to $7^{\frac{10}{2}}$.

• Use the property $\sqrt[n]{a}=a^{\frac{1}{n}}$ to express $\sqrt{7}$ as $7^{\frac{1}{2}}$.

• Then, raise it to the power of 10: ${\left(7^{\frac{1}{2}}\right)}^{10}$.

Step 1.1: Apply the exponent multiplication rule

Multiply the exponents together using $(a^m{)}^n=a^{mn}$.

• Multiply $\frac{1}{2}$ by 10: $7^{\frac{1}{2}\cdot 10}$.

Step 1.2: Simplify the exponent

Reduce the fraction in the exponent.

• Simplify $\frac{1}{2}\cdot 10$ to get $7^5$.

Step 1.3: Multiply by the coefficient

Multiply the simplified expression by 2.

• Compute $2\cdot 7^5$.

Step 2: Calculate the final result

Evaluate the expression.

• First, calculate $7^5$.

• Then, multiply the result by 2 to get the final answer.

Knowledge Notes:

To solve the given problem, several mathematical properties and rules were used:

1. Radical to Exponent Form: The square root of a number can be expressed as that number raised to the power of $\frac{1}{2}$, i.e., $\sqrt{a}=a^{\frac{1}{2}}$.

2. Power Rule: When an exponent is raised to another exponent, you multiply the exponents together, which is expressed as $(a^m{)}^n=a^{mn}$.

3. Simplification of Fractions: When multiplying a fraction by a whole number, you can simplify the calculation by multiplying the numerator by the whole number and keeping the denominator the same.

4. Multiplication: After simplifying the exponents, the final step is to multiply the coefficient (in this case, 2) by the result of the exponentiation.

5. Exponentiation: The process of raising a number to a power, which in this problem involves calculating $7^5$.

6. Arithmetic Operations: The final step involves basic arithmetic, multiplying two numbers together to get the final result.

By applying these rules and properties in a systematic manner, the problem can be solved efficiently and accurately.