Simplify The Expression: $-\sqrt{49} U^8 V^{12}$

$$

Understanding the Problem

When simplifying an expression, we need to break it down into its most basic components and then manipulate those components to arrive at the final result. In this case, we are given the expression $-\sqrt{49} u^8 v^{12}$ and we need to simplify it.

Breaking Down the Expression

The expression $-\sqrt{49} u^8 v^{12}$ can be broken down into several components:

• The negative sign $-$
• The square root of 49, denoted by $\sqrt{49}$
• The variables $u$ and $v$ raised to the powers of 8 and 12, respectively

Simplifying the Square Root

The square root of 49 can be simplified by finding the number that, when multiplied by itself, gives 49. In this case, the number is 7, since $7 \times 7 = 49$. Therefore, we can rewrite the expression as:

$$-\sqrt{49} u^8 v^{12} = -7 u^8 v^{12}$$

Simplifying the Variables

The variables $u$ and $v$ are raised to the powers of 8 and 12, respectively. We can simplify these variables by using the rules of exponents. Specifically, we can use the rule that states that when a variable is raised to a power, we can multiply the variable by itself as many times as the power indicates.

For example, $u^8$ can be rewritten as $u \times u \times u \times u \times u \times u \times u \times u$, and $v^{12}$ can be rewritten as $v \times v \times v \times v \times v \times v \times v \times v \times v \times v \times v \times v$.

However, instead of multiplying the variables, we can simply write them as $u^8$ and $v^{12}$.

Combining the Simplified Components

Now that we have simplified the square root and the variables, we can combine the simplified components to arrive at the final result.

$$-\sqrt{49} u^8 v^{12} = -7 u^8 v^{12}$$

Final Result

The final result of simplifying the expression $-\sqrt{49} u^8 v^{12}$ is $-7 u^8 v^{12}$.

Conclusion

Simplifying an expression involves breaking it down into its most basic components and then manipulating those components to arrive at the final result. In this case, we simplified the expression $-\sqrt{49} u^8 v^{12}$ by finding the square root of 49, simplifying the variables, and combining the simplified components.

Tips and Tricks

• When simplifying an expression, it's often helpful to break it down into its most basic components.
• Use the rules of exponents to simplify variables raised to powers.
• Combine the simplified components to arrive at the final result.

Common Mistakes to Avoid

• Failing to simplify the square root of a number.
• Not using the rules of exponents to simplify variables raised to powers.
• Not combining the simplified components to arrive at the final result.

Real-World Applications

Simplifying expressions is an important skill in mathematics and has many real-world applications. For example, in physics, simplifying expressions is used to describe the motion of objects and the behavior of physical systems. In engineering, simplifying expressions is used to design and optimize systems.

Practice Problems

• Simplify the expression $\sqrt{16} x^4 y^6$
• Simplify the expression $-\sqrt{81} u^2 v^3$
• Simplify the expression $\sqrt{25} x^3 y^2$

Solutions to Practice Problems

• $\sqrt{16} x^4 y^6 = 4 x^4 y^6$
• $-\sqrt{81} u^2 v^3 = -9 u^2 v^3$
• $\sqrt{25} x^3 y^2 = 5 x^3 y^2$

Conclusion

Simplifying expressions is an important skill in mathematics that has many real-world applications. By breaking down expressions into their most basic components and using the rules of exponents, we can simplify expressions and arrive at the final result.

$$

Understanding the Problem

When simplifying an expression, we need to break it down into its most basic components and then manipulate those components to arrive at the final result. In this case, we are given the expression $-\sqrt{49} u^8 v^{12}$ and we need to simplify it.

Q&A

Q: What is the square root of 49?

A: The square root of 49 is 7, since $7 \times 7 = 49$.

Q: How do we simplify the variables $u^8$ and $v^{12}$?

A: We can simplify the variables $u^8$ and $v^{12}$ by using the rules of exponents. Specifically, we can use the rule that states that when a variable is raised to a power, we can multiply the variable by itself as many times as the power indicates.

Q: What is the final result of simplifying the expression $-\sqrt{49} u^8 v^{12}$?

A: The final result of simplifying the expression $-\sqrt{49} u^8 v^{12}$ is $-7 u^8 v^{12}$.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Failing to simplify the square root of a number.
• Not using the rules of exponents to simplify variables raised to powers.
• Not combining the simplified components to arrive at the final result.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions is an important skill in mathematics that has many real-world applications. For example, in physics, simplifying expressions is used to describe the motion of objects and the behavior of physical systems. In engineering, simplifying expressions is used to design and optimize systems.

Q: How do we practice simplifying expressions?

A: We can practice simplifying expressions by working on practice problems, such as:

• Simplify the expression $\sqrt{16} x^4 y^6$
• Simplify the expression $-\sqrt{81} u^2 v^3$
• Simplify the expression $\sqrt{25} x^3 y^2$

Solutions to Practice Problems

• $\sqrt{16} x^4 y^6 = 4 x^4 y^6$
• $-\sqrt{81} u^2 v^3 = -9 u^2 v^3$
• $\sqrt{25} x^3 y^2 = 5 x^3 y^2$

Conclusion

Simplifying expressions is an important skill in mathematics that has many real-world applications. By breaking down expressions into their most basic components and using the rules of exponents, we can simplify expressions and arrive at the final result.