Technical MathematicsTerm 1 Assignment 2025 Grade 10---QUESTION 33.1 Simplify The Following Without The Use Of A Calculator. Leave Your Answer As A Positive Exponent:- 3.1.1 \[$\quad \frac{2^8}{2^6}\$\]- 3.1.2 \[$\frac{x^5 \times

Feb 26, 2025 by ADMIN 231 views

QUESTION 33.1 Simplify the following without the use of a calculator. Leave your answer as a positive exponent:

3.1.1 Simplifying Exponents with the Same Base

When simplifying exponents with the same base, we can use the quotient rule, which states that:

a^m / a^n = a^(m-n)

where a is the base and m and n are the exponents.

Let's apply this rule to the given expression:

$\frac{2^8}{2^6}$

Using the quotient rule, we can rewrite the expression as:

2^(8-6) = 2^2

Therefore, the simplified expression is:

2^2

3.1.2 Simplifying Exponents with Variables

Now, let's consider the expression:

$\frac{x^5 \times x^3}{x^2}$

To simplify this expression, we can use the product rule, which states that:

a^m \times a^n = a^(m+n)

where a is the base and m and n are the exponents.

We can also use the quotient rule, which states that:

a^m / a^n = a^(m-n)

where a is the base and m and n are the exponents.

Let's apply these rules to the given expression:

$\frac{x^5 \times x^3}{x^2}$

Using the product rule, we can rewrite the numerator as:

x^(5+3) = x^8

Now, we can use the quotient rule to simplify the expression:

x^8 / x^2 = x^(8-2) = x^6

Therefore, the simplified expression is:

x^6

3.1.3 Simplifying Exponents with Negative Exponents

Now, let's consider the expression:

$\frac{1}{2^6}$

To simplify this expression, we can use the rule for negative exponents, which states that:

a^(-n) = 1/a^n

where a is the base and n is the exponent.

Let's apply this rule to the given expression:

$\frac{1}{2^6}$

Using the rule for negative exponents, we can rewrite the expression as:

2^(-6)

Therefore, the simplified expression is:

2^(-6)

3.1.4 Simplifying Exponents with Zero Exponents

Now, let's consider the expression:

$\frac{2^8}{2^8}$

To simplify this expression, we can use the rule for zero exponents, which states that:

a^0 = 1

where a is the base.

Let's apply this rule to the given expression:

$\frac{2^8}{2^8}$

Using the rule for zero exponents, we can rewrite the expression as:

Therefore, the simplified expression is:

3.1.5 Simplifying Exponents with Fractional Exponents

Now, let's consider the expression:

$\frac{2^{3/2}}{2^{1/2}}$

To simplify this expression, we can use the quotient rule, which states that:

a^m / a^n = a^(m-n)

where a is the base and m and n are the exponents.

Let's apply this rule to the given expression:

$\frac{2^{3/2}}{2^{1/2}}$

Using the quotient rule, we can rewrite the expression as:

2^((3/2)-(1/2)) = 2^1

Therefore, the simplified expression is:

2^1

3.1.6 Simplifying Exponents with Mixed Exponents

Now, let's consider the expression:

$\frac{2^8 \times 2^3}{2^2}$

To simplify this expression, we can use the product rule, which states that:

a^m \times a^n = a^(m+n)

where a is the base and m and n are the exponents.

We can also use the quotient rule, which states that:

a^m / a^n = a^(m-n)

where a is the base and m and n are the exponents.

Let's apply these rules to the given expression:

$\frac{2^8 \times 2^3}{2^2}$

Using the product rule, we can rewrite the numerator as:

2^(8+3) = 2^11

Now, we can use the quotient rule to simplify the expression:

2^11 / 2^2 = 2^(11-2) = 2^9

Therefore, the simplified expression is:

2^9

Conclusion

In this article, we have simplified various expressions involving exponents, including expressions with the same base, variables, negative exponents, zero exponents, fractional exponents, and mixed exponents. We have used the quotient rule, product rule, and rules for negative exponents, zero exponents, and fractional exponents to simplify these expressions. By applying these rules, we have obtained simplified expressions in the form of positive exponents.

Key Takeaways

The quotient rule states that a^m / a^n = a^(m-n), where a is the base and m and n are the exponents.
The product rule states that a^m \times a^n = a^(m+n), where a is the base and m and n are the exponents.
The rule for negative exponents states that a^(-n) = 1/a^n, where a is the base and n is the exponent.
The rule for zero exponents states that a^0 = 1, where a is the base.
The rule for fractional exponents states that a^(m/n) = (a^m)(1/n), where a is the base and m and n are the exponents.

Practice Problems

Simplify the expression: $\frac{2^8}{2^4}$
Simplify the expression: $\frac{x^5 \times x^2}{x^3}$
Simplify the expression: $\frac{1}{2^8}$
Simplify the expression: $\frac{2^8}{2^8}$
Simplify the expression: $\frac{2^{3/2}}{2^{1/2}}$

Solutions

2^4
x^4
2^(-8)
1
2^1
Technical Mathematics Term 1 Assignment 2025 Grade 10 ===========================================================

QUESTION 33.1 Simplify the following without the use of a calculator. Leave your answer as a positive exponent:

Q&A: Simplifying Exponents

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that a^m / a^n = a^(m-n), where a is the base and m and n are the exponents.

Q: How do I simplify an expression with the same base and different exponents?

A: To simplify an expression with the same base and different exponents, you can use the quotient rule. For example, if you have the expression 2^8 / 2^6, you can simplify it by subtracting the exponents: 2^(8-6) = 2^2.

Q: What is the product rule for exponents?

A: The product rule for exponents states that a^m \times a^n = a^(m+n), where a is the base and m and n are the exponents.

Q: How do I simplify an expression with variables and exponents?

A: To simplify an expression with variables and exponents, you can use the product rule and the quotient rule. For example, if you have the expression (x^5 \times x^3) / x^2, you can simplify it by first using the product rule to combine the exponents: x^(5+3) = x^8, and then using the quotient rule to subtract the exponents: x^8 / x^2 = x^(8-2) = x^6.

Q: What is the rule for negative exponents?

A: The rule for negative exponents states that a^(-n) = 1/a^n, where a is the base and n is the exponent.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can use the rule for negative exponents. For example, if you have the expression 1/2^6, you can simplify it by rewriting it as 2^(-6).

Q: What is the rule for zero exponents?

A: The rule for zero exponents states that a^0 = 1, where a is the base.

Q: How do I simplify an expression with a zero exponent?

A: To simplify an expression with a zero exponent, you can use the rule for zero exponents. For example, if you have the expression 2^8 / 2^8, you can simplify it by rewriting it as 1.

Q: What is the rule for fractional exponents?

A: The rule for fractional exponents states that a^(m/n) = (a^m)(1/n), where a is the base and m and n are the exponents.

Q: How do I simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, you can use the rule for fractional exponents. For example, if you have the expression 2^(3/2) / 2^(1/2), you can simplify it by first rewriting it as (2³⁾(1/2) / 2^(1/2), and then simplifying it further by using the quotient rule: (2³⁾(1/2) / 2^(1/2) = 2^(3/2-1/2) = 2^1.

Practice Problems

Simplify the expression: $\frac{2^8}{2^4}$
Simplify the expression: $\frac{x^5 \times x^2}{x^3}$
Simplify the expression: $\frac{1}{2^8}$
Simplify the expression: $\frac{2^8}{2^8}$
Simplify the expression: $\frac{2^{3/2}}{2^{1/2}}$

Solutions

2^4
x^4
2^(-8)
1
2^1

Additional Resources

For more practice problems and solutions, visit our website at [insert website URL].
For additional resources and study guides, visit our online store at [insert online store URL].
For one-on-one tutoring and support, contact us at [insert contact information].