Simplify: 3.1.1 M; 3.1.2 (M¹⁵)⁵

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QUESTION 2

2.1.1 What will be prime factors of 200?

Find the prime factorization of 200.
$200 = 2 \times 100$ $200 = 2 \times 2 \times 50$ $200 = 2 \times 2 \times 2 \times 25$ $200 = 2 \times 2 \times 2 \times 5 \times 5$ $200 = 2^3 \times 5^2$
The prime factors of 200 are 2 and 5.
The final answer is $\boxed{\text{2 and 5}}$

2.1.2 Hence, find HCF of 100 and 200 factors.
Write the prime factorization of 100 and 200.
Given $100 = 2^2 \times 5^2$ .
From 2.1.1, $200 = 2^3 \times 5^2$ .
Identify the common prime factors with the lowest powers.
The common prime factors are 2 and 5.
The lowest power of 2 is $2^2$ .
The lowest power of 5 is $5^2$ .
Calculate the HCF.
$\text{HCF}(100, 200) = 2^2 \times 5^2 = 4 \times 25 = 100$
The final answer is $\boxed{\text{100}}$

2.2 John and Anna, share a total profit of R $315$ in the ratio $3:4$ . Calculate how much money Anna will receive?
Determine the total number of parts in the ratio.
Total parts = $3 + 4 = 7$ .
Calculate Anna's share of the profit.
Anna's share is 4 parts out of 7.
$\text{Anna's share} = \frac{4}{7} \times \text{R}315$ $\text{Anna's share} = 4 \times \frac{315}{7}$ $\text{Anna's share} = 4 \times 45$ $\text{Anna's share} = \text{R}180$
The final answer is $\boxed{\text{R180}}$

2.3 A local supermarket sells tomatoes for:
- 3 kg bag costs R $70$
- 2 kg bag costs R $50$ Calculate the price per kg for each option, which option is expensive.
Calculate the price per kg for the 3 kg bag.
$\text{Price per kg (3 kg bag)} = \frac{\text{R}70}{3 \text{ kg}} \approx \text{R}23.33 \text{ per kg}$
Calculate the price per kg for the 2 kg bag.
$\text{Price per kg (2 kg bag)} = \frac{\text{R}50}{2 \text{ kg}} = \text{R}25.00 \text{ per kg}$
Compare the prices to determine which option is more expensive.
R $25.00$ is greater than R $23.33$ .
The 2 kg bag is more expensive per kg.
The final answer is $\boxed{\text{3 kg bag: R23.33/kg, 2 kg bag: R25.00/kg. The 2 kg bag is more expensive.}}$

2.4 Mr Khumalo invests R $2000$ in a savings account at a financial institution that offers a simple interest rate of $4\%$ per annum. Calculate the total value of her investment over a period of 3 years.
Identify the given values.
Principal ( $P$ ) = R $2000$ .
Interest rate ( $R$ ) = $4\% = 0.04$ .
Time ( $T$ ) = 3 years.
Calculate the simple interest ( $I$ ).
$I = P \times R \times T$ $I = 2000 \times 0.04 \times 3$ $I = 80 \times 3$ $I = \text{R}240$
Calculate the total value of the investment ( $A$ ).
$A = P + I$ $A = 2000 + 240$ $A = \text{R}2240$
The final answer is $\boxed{\text{R2240}}$

QUESTION 3

3.1.1 Simplify $M^x \cdot M^y$ .
Apply the rule for multiplying exponents with the same base
$a^m \cdot a^n = a^{m+n}$ .
$M^x \cdot M^y = M^{x+y}$
The final answer is $\boxed{M^{x+y}}$

3.1.2 Simplify $(M^x)^y$ .
Apply the rule for a power of a power
$(a^m)^n = a^{m \cdot n}$ .
$(M^x)^y = M^{x \cdot y}$
The final answer is $\boxed{M^{xy}}$

3.2 Simplify $\sqrt{125} + (\frac{2}{3})^0 - (2 \div 0.2)$ .
Simplify $\sqrt{125}$ .
$\sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5}$
Simplify $(\frac{2}{3})^0$ .
Any non-zero number raised to the power of 0 is 1.
$(\frac{2}{3})^0 = 1$
Simplify $(2 \div 0.2)$ .
$2 \div 0.2 = 2 \div \frac{2}{10} = 2 \times \frac{10}{2} = 10$
Combine the simplified terms.
$5\sqrt{5} + 1 - 10$ $5\sqrt{5} - 9$
The final answer is $\boxed{5\sqrt{5} - 9}$

3.3 Simplify expression using laws exponents. $\frac{(x^2y^3)^3}{x^5y^5}$ .
Apply the power of a product rule $(ab)^n = a^n b^n$ and power of a power rule $(a^m)^n = a^{mn}$ to the numerator.
$(x^2y^3)^3 = (x^2)^3 (y^3)^3 = x^{2 \times 3} y^{3 \times 3} = x^6 y^9$
Rewrite the expression with the simplified numerator.
$\frac{x^6 y^9}{x^5 y^5}$
Apply the division rule for exponents $\frac{a^m}{a^n} = a^{m-n}$ .
$x^{6-5} y^{9-5}$ $x^1 y^4$ $xy^4$
The final answer is $\boxed{xy^4}$

3.4 A scientist estimates that a water sample contains approximately 50,700,000 microorganisms. Express this number of bacteria in scientific notation.
Move the decimal point to the left until there is only one non-zero digit to its left.
The number is 50,700,000. The decimal point is initially after the last 0.
Move it 7 places to the left to get 5.07.
Count the number of places the decimal point was moved.
This number will be the exponent of 10. Since the original number is greater than 1, the exponent is positive.
The decimal point was moved 7 places.
$50,700,000 = 5.07 \times 10^7$
The final answer is $\boxed{5.07 \times 10^7}$

QUESTION 4

4.1 Simplify $3x^3 - 3x^2 + 4x^3 + 4x^2$ .
Group like terms together.
$(3x^3 + 4x^3) + (-3x^2 + 4x^2)$
Combine the coefficients of the like terms.
$(3+4)x^3 + (-3+4)x^2$ $7x^3 + 1x^2$ $7x^3 + x^2$
The final answer is $\boxed{7x^3 + x^2}$

4.2 Given: $x + x^6 + \frac{x}{2} - 2 - 4x^2$ .
4.2.1 Write down the value of the constant term.
The constant term is the term that does not contain any variables.
In the expression, the constant term is $-2$ .
The final answer is $\boxed{-2}$

4.2.2 State the coefficient of $x^3$ .
The coefficient of $x^3$ is the numerical factor multiplying $x^3$ .
In the given expression, there is no $x^3$ term. This means its coefficient is 0.
The final answer is $\boxed{0}$

4.2.3 What is the degree of polynomial?
The degree of a polynomial is the highest exponent of the variable in the polynomial.
The terms in the polynomial are $x^1, x^6, x^1, x^0, x^2$ .
The exponents are $1, 6, 1, 0, 2$ .
The highest exponent is 6.
The final answer is $\boxed{6}$

4.3 Simplify: $4(-x^2 + x - 2) - 4x$ .
Distribute the 4 into the parenthesis.
$4(-x^2) + 4(x) + 4(-2) - 4x$ $-4x^2 + 4x - 8 - 4x$
Combine like terms.
$-4x^2 + (4x - 4x) - 8$ $-4x^2 + 0x - 8$ $-4x^2 - 8$
The final answer is $\boxed{-4x^2 - 8}$

4.4 Simplify: $\frac{18x^2y^3 + 6x^3y^2 - 12x^2y}{3xy^2}$ .
Divide each term in the numerator by the denominator.
$\frac{18x^2y^3}{3xy^2} + \frac{6x^3y^2}{3xy^2} - \frac{12x^2y}{3xy^2}$
Simplify each term using exponent rules ( $\frac{a^m}{a^n} = a^{m-n}$ ).
For the first term:
$\frac{18}{3} x^{2-1} y^{3-2} = 6xy^1 = 6xy$
For the second term:
$\frac{6}{3} x^{3-1} y^{2-2} = 2x^2y^0 = 2x^2(1) = 2x^2$
For the third term:
$\frac{-12}{3} x^{2-1} y^{1-2} = -4xy^{-1} = -\frac{4x}{y}$
Combine the simplified terms.
$6xy + 2x^2 - \frac{4x}{y}$
The final answer is $\boxed{6xy + 2x^2 - \frac{4x}{y}}$

QUESTION 5

5.1 Consider: $7; 11; 15; 19; ...$
5.1.1 Write down next two terms.
Find the common difference ( $d$ ) between consecutive terms.
$d = 11 - 7 = 4$ $d = 15 - 11 = 4$ $d = 19 - 15 = 4$
The common difference is 4.
Add the common difference to the last given term to find the next terms.
The $5^{th}$ term: $19 + 4 = 23$ .
The $6^{th}$ term: $23 + 4 = 27$ .
The final answer is $\boxed{23; 27}$

5.1.2 Find the general rule for the pattern ( $T_n$ ).
Use the formula for the $n^{th}$ term of an arithmetic sequence
$T_n = a + (n-1)d$ .
Here, $a$ (first term) $= 7$ and $d$ (common difference) $= 4$ .
Substitute the values into the formula.
$T_n = 7 + (n-1)4$ $T_n = 7 + 4n - 4$ $T_n = 4n + 3$
The final answer is $\boxed{T_n = 4n + 3}$

5.2 Solve for $x$ : $3 - 3x = -15$ .
Subtract 3 from both sides of the equation.
$3 - 3x - 3 = -15 - 3$ $-3x = -18$
Divide both sides by $-3$ .
$\frac{-3x}{-3} = \frac{-18}{-3}$ $x = 6$
The final answer is $\boxed{x = 6}$

5.3 When a certain number is divided by 4 and then decreased by 3, the result is equal to the same number divided by 2 and then decreased by 9. Set up an algebraic equation to describe the problem and solve the equation.
Let the certain number be $y$ .
Translate the first part of the problem into an algebraic expression
"a certain number is divided by 4 and then decreased by 3".
$\frac{y}{4} - 3$
Translate the second part of the problem into an algebraic expression
"the same number divided by 2 and then decreased by 9".
$\frac{y}{2} - 9$
Set up the equation by equating the two expressions.
$\frac{y}{4} - 3 = \frac{y}{2} - 9$
Solve the equation for $y$ .
Multiply the entire equation by the least common multiple (LCM) of the denominators (4 and 2), which is 4, to eliminate fractions.
$4 \left( \frac{y}{4} - 3 \right) = 4 \left( \frac{y}{2} - 9 \right)$ $4 \cdot \frac{y}{4} - 4 \cdot 3 = 4 \cdot \frac{y}{2} - 4 \cdot 9$ $y - 12 = 2y - 36$
Gather $y$ terms on one side and constant terms on the other.
Subtract $y$ from both sides:
$-12 = 2y - y - 36$ $-12 = y - 36$
Add 36 to both sides:
$-12 + 36 = y$ $y = 24$
The final answer is $\boxed{\text{Equation: } \frac{y}{4} - 3 = \frac{y}{2} - 9, \text{ Solution: } y = 24}$

Simplify: 3.1.1 M; 3.1.2 (M¹⁵)⁵

Step-by-step solution

Find the prime factorization of 200.

Write the prime factorization of 100 and 200.

Identify the common prime factors with the lowest powers.

Calculate the HCF.

Determine the total number of parts in the ratio.

Calculate Anna's share of the profit.

Calculate the price per kg for the 3 kg bag.

Calculate the price per kg for the 2 kg bag.

Compare the prices to determine which option is more expensive.

Identify the given values.

Calculate the simple interest (III).

Calculate the total value of the investment (AAA).

Apply the rule for multiplying exponents with the same base

Apply the rule for a power of a power

Simplify 125\sqrt{125}125​.

Simplify (23)0(\frac{2}{3})^0(32​)0.

Simplify (2÷0.2)(2 \div 0.2)(2÷0.2).

Combine the simplified terms.

Apply the power of a product rule (ab)n=anbn(ab)^n = a^n b^n(ab)n=anbn and power of a power rule (am)n=amn(a^m)^n = a^{mn}(am)n=amn to the numerator.

Rewrite the expression with the simplified numerator.

Apply the division rule for exponents aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n.

Move the decimal point to the left until there is only one non-zero digit to its left.

Count the number of places the decimal point was moved.

Group like terms together.

Combine the coefficients of the like terms.

Distribute the 4 into the parenthesis.

Combine like terms.

Divide each term in the numerator by the denominator.

Simplify each term using exponent rules (aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n).

Combine the simplified terms.

Find the common difference (ddd) between consecutive terms.

Add the common difference to the last given term to find the next terms.

Use the formula for the nthn^{th}nth term of an arithmetic sequence

Substitute the values into the formula.

Subtract 3 from both sides of the equation.

Divide both sides by −3-3−3.

Let the certain number be yyy.

Translate the first part of the problem into an algebraic expression

Translate the second part of the problem into an algebraic expression

Set up the equation by equating the two expressions.

Solve the equation for yyy.

Gather yyy terms on one side and constant terms on the other.

Still stuck on this one?

More Mathematics Questions

Calculate the simple interest ( $I$ ).

Calculate the total value of the investment ( $A$ ).

Simplify $\sqrt{125}$ .

Simplify $(\frac{2}{3})^0$ .

Simplify $(2 \div 0.2)$ .

Apply the power of a product rule $(ab)^n = a^n b^n$ and power of a power rule $(a^m)^n = a^{mn}$ to the numerator.

Apply the division rule for exponents $\frac{a^m}{a^n} = a^{m-n}$ .

Simplify each term using exponent rules ( $\frac{a^m}{a^n} = a^{m-n}$ ).

Find the common difference ( $d$ ) between consecutive terms.

Use the formula for the $n^{th}$ term of an arithmetic sequence

Divide both sides by $-3$ .

Let the certain number be $y$ .

Solve the equation for $y$ .

Gather $y$ terms on one side and constant terms on the other.