QUESTION 6 [EXPONENTS] Calculate the values of the following: 6.1 2/60, 6.2 53 × 54, 6.3 85 + 83, 6.4 Show by…

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QUESTION 6 [EXPONENTS] Calculate the values of the following: 6.1 2/60, 6.2 53 × 54, 6.3 85 + 83, 6.4 Show by means of calculations that 25 is bigger than 52.

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QUESTION 6 [EXPONENTS]

6.1.

Evaluate the term with the exponent.
Any non-zero number raised to the power of 0 is 1.
$6^0 = 1$
Substitute this value back into the expression and simplify.
$\frac{2}{1} = 2$
The value is $\boxed{2}$ .

6.2.
Apply the rule for multiplying exponents with the same base
$a^m \times a^n = a^{m+n}$ .
$5^3 \times 5^4 = 5^{3+4}$
Add the exponents and calculate the final value.
$5^7 = 78125$
The value is $\boxed{78125}$ .

6.3.
Apply the rule for dividing exponents with the same base
$a^m \div a^n = a^{m-n}$ .
$8^5 \div 8^3 = 8^{5-3}$
Subtract the exponents and calculate the final value.
$8^2 = 64$
The value is $\boxed{64}$ .

6.4.
Calculate the value of $2^5$ .
$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$
Calculate the value of $5^2$ .
$5^2 = 5 \times 5 = 25$
Compare the two values.
$32 > 25$
Therefore, $2^5$ is bigger than $5^2$ .

QUESTION 7 [NUMERIC AND GEOMETRIC PATTERNS]

7.1.1.
Find the common difference between consecutive terms.
$-5 - (-2) = -3$ $-8 - (-5) = -3$ $-11 - (-8) = -3$
Add the common difference to the last given term to find the next term.
$-11 + (-3) = -14$
The next term is $\boxed{-14}$ .

7.1.2.
Identify the first term ( $a$ ) and the common difference ( $d$ ).
$a = -2$ $d = -3$
Use the general formula for an arithmetic sequence, $T_n = a + (n-1)d$ .
$T_n = -2 + (n-1)(-3)$
Simplify the expression to find the general rule.
$T_n = -2 - 3n + 3$ $T_n = 1 - 3n$
The general rule is $\boxed{T_n = 1 - 3n}$ .

7.1.3.
Use the general rule $T_n = 1 - 3n$ found in 7.1.2.
Substitute $n = 50$ into the rule to find the 50th term.
$T_{50} = 1 - 3(50)$
Perform the multiplication and subtraction.
$T_{50} = 1 - 150$ $T_{50} = -149$
The 50th term is $\boxed{-149}$ .

QUESTION 6 [EXPONENTS] Calculate the values of the following: 6.1 2/60, 6.2 53 x 54, 6.3 85 + 83, 6.4 Show by means of calculations that 25 is bigger than 52.

Step-by-step solution

Evaluate the term with the exponent.

Substitute this value back into the expression and simplify.

Apply the rule for multiplying exponents with the same base

Add the exponents and calculate the final value.

Apply the rule for dividing exponents with the same base

Subtract the exponents and calculate the final value.

Calculate the value of $2^5$ .

Calculate the value of $5^2$ .

Compare the two values.

Find the common difference between consecutive terms.

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

Identify the first term ( $a$ ) and the common difference ( $d$ ).

Use the general formula for an arithmetic sequence, $T_n = a + (n-1)d$ .

Simplify the expression to find the general rule.

Use the general rule $T_n = 1 - 3n$ found in 7.1.2.

Substitute $n = 50$ into the rule to find the 50th term.

Perform the multiplication and subtraction.

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