Solve: $4^2 - \sqrt{2} T$2. Find The Quotient Of The L.C.M. And H.C.F. Of 12 And 18.3. By How Much Is 4 3 7 4 \frac{3}{7} 4 7 3 ​ Greater Than 4 5 \frac{4}{5} 5 4 ​ ?4. Find The Cost Of 14 Packets Of Powdered Soap At $0.50$ Per

Problem 1: Solving a Quadratic Equation with a Square Root

Understanding the Problem

We are given a quadratic equation in the form of $4^2 - \sqrt{2} t$. Our goal is to solve for the value of $t$.

Breaking Down the Problem

To solve this equation, we need to isolate the variable $t$. We can start by evaluating the expression $4^2$, which is equal to $16$. So, the equation becomes $16 - \sqrt{2} t$.

Solving for $t$

To solve for $t$, we need to get rid of the square root term. We can do this by multiplying both sides of the equation by $\sqrt{2}$. This gives us:

$$16\sqrt{2} - 2t = 0$$

Isolating $t$

Now, we can isolate $t$ by dividing both sides of the equation by $-2$. This gives us:

$$t = \frac{16\sqrt{2}}{2}$$

Simplifying the Expression

We can simplify the expression by dividing the numerator and denominator by $2$. This gives us:

$$t = 8\sqrt{2}$$

Problem 2: Finding the L.C.M. and H.C.F. of Two Numbers

Understanding the Problem

We are given two numbers, $12$ and $18$. Our goal is to find the least common multiple (L.C.M.) and the highest common factor (H.C.F.) of these two numbers.

Breaking Down the Problem

To find the L.C.M. and H.C.F., we need to first find the prime factors of both numbers. The prime factors of $12$ are $2^2 \cdot 3$, and the prime factors of $18$ are $2 \cdot 3^2$.

Finding the L.C.M.

The L.C.M. is the product of the highest powers of all the prime factors. In this case, the L.C.M. is $2^2 \cdot 3^2 = 36$.

Finding the H.C.F.

The H.C.F. is the product of the lowest powers of all the prime factors. In this case, the H.C.F. is $2 \cdot 3 = 6$.

Finding the Quotient of the L.C.M. and H.C.F.

To find the quotient of the L.C.M. and H.C.F., we can divide the L.C.M. by the H.C.F. This gives us:

$$\frac{36}{6} = 6$$

Problem 3: Comparing Two Fractions

Understanding the Problem

We are given two fractions, $4 \frac{3}{7}$ and $\frac{4}{5}$. Our goal is to find the difference between these two fractions.

Breaking Down the Problem

To compare these fractions, we need to convert them to equivalent fractions with the same denominator. The least common multiple of $7$ and $5$ is $35$, so we can convert both fractions to have a denominator of $35$.

Converting the Fractions

The equivalent fraction for $4 \frac{3}{7}$ is $\frac{31}{7} = \frac{31 \cdot 5}{7 \cdot 5} = \frac{155}{35}$.

The equivalent fraction for $\frac{4}{5}$ is $\frac{4 \cdot 7}{5 \cdot 7} = \frac{28}{35}$.

Finding the Difference

To find the difference between these two fractions, we can subtract the second fraction from the first fraction. This gives us:

$$\frac{155}{35} - \frac{28}{35} = \frac{127}{35}$$

Simplifying the Fraction

We can simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is $1$. This gives us:

$$\frac{127}{35}$$

Problem 4: Finding the Cost of Powdered Soap

Understanding the Problem

We are given the cost of powdered soap at $0.50 per packet. Our goal is to find the cost of 14 packets of powdered soap.

Breaking Down the Problem

To find the cost of 14 packets of powdered soap, we can multiply the cost per packet by the number of packets. This gives us:

$$0.50 \cdot 14 = 7$$

Finding the Total Cost

The total cost is equal to the cost per packet multiplied by the number of packets. In this case, the total cost is $7.

The final answer is: $\boxed{7}$

Understanding Mathematical Concepts

Mathematics is a vast and complex subject that encompasses various branches, including algebra, geometry, trigonometry, and calculus. It is a fundamental tool for problem-solving, critical thinking, and logical reasoning. In this article, we will address some common mathematical problems and provide step-by-step solutions to help clarify the concepts.

Q&A Session

Q: What is the value of x in the equation 2x + 5 = 11?

A: To solve for x, we need to isolate the variable x. We can do this by subtracting 5 from both sides of the equation and then dividing both sides by 2.

2x + 5 - 5 = 11 - 5 2x = 6 x = 6/2 x = 3

Q: What is the least common multiple (LCM) of 8 and 12?

A: To find the LCM, we need to list the multiples of both numbers and find the smallest multiple that is common to both.

Multiples of 8: 8, 16, 24, 32, ... Multiples of 12: 12, 24, 36, 48, ...

The smallest multiple that is common to both is 24. Therefore, the LCM of 8 and 12 is 24.

Q: What is the value of y in the equation y/4 = 9?

A: To solve for y, we need to isolate the variable y. We can do this by multiplying both sides of the equation by 4.

y/4 = 9 y = 9 * 4 y = 36

Q: What is the difference between the sum of the interior angles of a triangle and the sum of the interior angles of a quadrilateral?

A: The sum of the interior angles of a triangle is 180 degrees, and the sum of the interior angles of a quadrilateral is 360 degrees. Therefore, the difference between the two is:

360 - 180 = 180

Q: What is the value of z in the equation z - 2 = 7?

A: To solve for z, we need to isolate the variable z. We can do this by adding 2 to both sides of the equation.

z - 2 + 2 = 7 + 2 z = 9

Q: What is the product of the factors of 12?

A: The factors of 12 are 1, 2, 3, 4, 6, and 12. Therefore, the product of the factors is:

1 * 2 * 3 * 4 * 6 * 12 = 1728

Q: What is the value of w in the equation w/3 = 12?

A: To solve for w, we need to isolate the variable w. We can do this by multiplying both sides of the equation by 3.

w/3 = 12 w = 12 * 3 w = 36

Conclusion

Mathematics is a subject that requires practice and patience to master. By solving problems and clarifying concepts, we can gain a deeper understanding of mathematical principles and develop problem-solving skills. In this article, we have addressed some common mathematical problems and provided step-by-step solutions to help clarify the concepts.

The final answer is: $\boxed{36}$