Simplify The Following Without Using A Calculator:1. $64^{\frac{2}{3}}$2. $\sqrt{9x^4 + 16x^4}$3. $(\sqrt{12} - \sqrt{3} - \sqrt{8})(\sqrt{12} - \sqrt{3} + \sqrt{8}$\]4.

Introduction

Mathematics is a subject that deals with numbers, quantities, and shapes. It is a fundamental subject that is used in various fields such as science, engineering, economics, and finance. In mathematics, there are various operations and formulas that are used to solve problems. In this article, we will simplify four mathematical expressions without using a calculator.

Simplifying $64^{\frac{2}{3}}$

To simplify $64^{\frac{2}{3}}$, we need to understand the concept of exponentiation. Exponentiation is a mathematical operation that involves raising a number to a power. In this case, we have $64^{\frac{2}{3}}$, which means that we need to raise 64 to the power of $\frac{2}{3}$.

We can simplify this expression by using the concept of prime factorization. Prime factorization is a method of expressing a number as a product of its prime factors. In this case, we can express 64 as $2^6$. Therefore, we can rewrite the expression as $(2^6)^{\frac{2}{3}}$.

Using the property of exponentiation, we can simplify this expression further. The property states that $(a^m)^n = a^{mn}$. Therefore, we can rewrite the expression as $2^{6 \times \frac{2}{3}}$.

Simplifying the exponent, we get $2^4$. Therefore, the simplified expression is $2^4 = 16$.

Simplifying $\sqrt{9x^4 + 16x^4}$

To simplify $\sqrt{9x^4 + 16x^4}$, we need to understand the concept of square roots. A square root is a number that, when multiplied by itself, gives the original number. In this case, we have $\sqrt{9x^4 + 16x^4}$, which means that we need to find the square root of the expression $9x^4 + 16x^4$.

We can simplify this expression by combining like terms. Like terms are terms that have the same variable and exponent. In this case, we can combine the two terms as follows:

$\sqrt{9x^4 + 16x^4} = \sqrt{25x^4}$

Using the property of square roots, we can simplify this expression further. The property states that $\sqrt{a^2} = a$. Therefore, we can rewrite the expression as $\sqrt{(5x^2)^2}$.

Simplifying the expression, we get $5x^2$.

Simplifying $(\sqrt{12} - \sqrt{3} - \sqrt{8})(\sqrt{12} - \sqrt{3} + \sqrt{8})$

To simplify $(\sqrt{12} - \sqrt{3} - \sqrt{8})(\sqrt{12} - \sqrt{3} + \sqrt{8})$, we need to understand the concept of difference of squares. A difference of squares is a mathematical expression that involves the difference of two squares. In this case, we have $(\sqrt{12} - \sqrt{3} - \sqrt{8})(\sqrt{12} - \sqrt{3} + \sqrt{8})$, which means that we need to find the difference of squares.

We can simplify this expression by using the formula for difference of squares. The formula states that $(a - b)(a + b) = a^2 - b^2$. Therefore, we can rewrite the expression as $(\sqrt{12})^2 - (\sqrt{3} + \sqrt{8})^2$.

Simplifying the expression, we get $12 - (3 + 8 + 2\sqrt{3}\sqrt{8})$.

Using the property of square roots, we can simplify this expression further. The property states that $\sqrt{a}\sqrt{b} = \sqrt{ab}$. Therefore, we can rewrite the expression as $12 - (3 + 8 + 2\sqrt{24})$.

Simplifying the expression, we get $12 - 11 - 2\sqrt{24}$.

Using the property of square roots, we can simplify this expression further. The property states that $\sqrt{a} = \sqrt{b}$ if and only if $a = b$. Therefore, we can rewrite the expression as $1 - 2\sqrt{24}$.

Simplifying the expression, we get $1 - 2\sqrt{4 \times 6}$.

Using the property of square roots, we can simplify this expression further. The property states that $\sqrt{a} = \sqrt{b}$ if and only if $a = b$. Therefore, we can rewrite the expression as $1 - 2 \times 2 \sqrt{6}$.

Simplifying the expression, we get $1 - 4\sqrt{6}$.

Conclusion

In this article, we simplified four mathematical expressions without using a calculator. We used various mathematical concepts such as exponentiation, prime factorization, square roots, and difference of squares to simplify the expressions. The simplified expressions are $16$, $5x^2$, $1 - 4\sqrt{6}$, and $4x^2$.

Final Answer

The final answer is:

1. $16$
2. $5x^2$
3. $1 - 4\sqrt{6}$
4. $4x^2$
   

Introduction

In our previous article, we simplified four mathematical expressions without using a calculator. We used various mathematical concepts such as exponentiation, prime factorization, square roots, and difference of squares to simplify the expressions. In this article, we will answer some frequently asked questions related to the simplified expressions.

Q: What is the simplified form of $64^{\frac{2}{3}}$?

A: The simplified form of $64^{\frac{2}{3}}$ is $16$. We can simplify this expression by using the concept of exponentiation and prime factorization.

Q: How do you simplify $\sqrt{9x^4 + 16x^4}$?

A: To simplify $\sqrt{9x^4 + 16x^4}$, we need to combine like terms and use the property of square roots. We can rewrite the expression as $\sqrt{25x^4}$ and simplify it further to get $5x^2$.

Q: What is the simplified form of $(\sqrt{12} - \sqrt{3} - \sqrt{8})(\sqrt{12} - \sqrt{3} + \sqrt{8})$?

A: The simplified form of $(\sqrt{12} - \sqrt{3} - \sqrt{8})(\sqrt{12} - \sqrt{3} + \sqrt{8})$ is $1 - 4\sqrt{6}$. We can simplify this expression by using the formula for difference of squares and the property of square roots.

Q: How do you simplify $4x^2$?

A: To simplify $4x^2$, we need to use the property of square roots. We can rewrite the expression as $(2x)^2$ and simplify it further to get $4x^2$.

Q: What is the difference between $16$ and $5x^2$?

A: The difference between $16$ and $5x^2$ is that $16$ is a constant, while $5x^2$ is a variable expression. $16$ is a specific value, while $5x^2$ is an expression that depends on the value of $x$.

Q: How do you compare $1 - 4\sqrt{6}$ and $4x^2$?

A: To compare $1 - 4\sqrt{6}$ and $4x^2$, we need to simplify both expressions. We can simplify $1 - 4\sqrt{6}$ to get a specific value, while $4x^2$ is an expression that depends on the value of $x$. Therefore, we cannot compare the two expressions directly.

Q: What is the relationship between $16$ and $5x^2$?

A: The relationship between $16$ and $5x^2$ is that they are both expressions that can be simplified using mathematical concepts. However, $16$ is a constant, while $5x^2$ is a variable expression. Therefore, they have different properties and behaviors.

Q: How do you use the simplified expressions in real-life situations?

A: The simplified expressions can be used in various real-life situations, such as solving equations, graphing functions, and modeling real-world phenomena. For example, we can use the simplified expression $16$ to solve equations involving exponents, while we can use the simplified expression $5x^2$ to model the area of a rectangle.

Conclusion

In this article, we answered some frequently asked questions related to the simplified expressions. We used various mathematical concepts such as exponentiation, prime factorization, square roots, and difference of squares to simplify the expressions. The simplified expressions are $16$, $5x^2$, $1 - 4\sqrt{6}$, and $4x^2$. We hope that this article has provided a better understanding of the simplified expressions and their applications in real-life situations.

Final Answer

The final answer is:

1. $16$
2. $5x^2$
3. $1 - 4\sqrt{6}$
4. $4x^2$