CBSE Class 7 Maths Chapter 11 Exponent NCERT Solutions

To solve this, we use the exponent rule $(a^m)^n = a^{m \times n}$. This rule states that when a power is raised to another power, we multiply the exponents.

Applying this rule to the given expression:

$$[(-3)^2]^3\; =\; (-3)^\{2\; \backslash times\; 3\}$$

$$=\; (-3)^6$$

Therefore, the expression $[(-3)^2]^3$ is equal to $(-3)^6$.

The correct option is (b).

We use the rule for dividing exponents with the same base, which states that $a^m \div a^n = a^{m-n}$. When dividing powers that have the same base, we subtract the exponent of the divisor from the exponent of the dividend.

Applying this rule to the given expression:

$$x^8\; \backslash div\; x^2\; =\; \backslash frac\{x^8\}\{x^2\}$$

$$=\; x^\{8-2\}$$

$$=\; x^6$$

Thus, $x^8 \div x^2$ is equal to $x^6$.

The correct option is (b).

First, let's identify the terms given in the question:

• The square of $x$ is $x^2$.
• The cube of $x$ is $x^3$.

Now, we need to find the product of these two terms. We use the rule for multiplying exponents with the same base, which states that $a^m \times a^n = a^{m+n}$.

Calculating the product:

$$x^2\; \backslash times\; x^3\; =\; x^\{2+3\}$$

$$=\; x^5$$

This result, $x^5$, represents the fifth power of $x$.

The correct option is (c).

We use the exponent rule for division where the exponents are the same but the bases are different: $a^m \div b^m = (a \div b)^m$. This rule allows us to combine terms with the same exponent by dividing their bases.

Applying this rule to the given expression:

$$x^5\; \backslash div\; y^5\; =\; \backslash left(\backslash frac\{x\}\{y\}\backslash right)^5$$

This can also be written as $(x \div y)^5$.

The correct option is (c).

This question involves the rule for multiplying exponential terms that have the same base. The rule states that when multiplying powers with the same base, we keep the base the same and add the exponents.

The rule is: $a^m \times a^n = a^{m+n}$.

Therefore, $a^m \times a^n$ is equal to $a^{m+n}$.

The correct option is (c).

We need to evaluate the expression $(1^0 + 2^0 + 3^0)$. A fundamental rule of exponents is that any non-zero number raised to the power of zero is equal to 1.

Applying this rule:

• $1^0 = 1$
• $2^0 = 1$
• $3^0 = 1$

Now, we sum these values:

1^0 + 2^0 + 3^0 = 1 + 1 + 1 = 3 So, the value of the expression is 3.

The correct option is (c).

To find the value of the expression $\frac{(10^{22} + 10^{20})}{10^{20}}$, we can simplify the numerator by factoring out the common term $10^{20}$.

We can rewrite $10^{22}$ as $10^{20} \times 10^2$.

So the expression becomes:

$$\backslash frac\{10^\{20\}\; \backslash times\; 10^2\; +\; 10^\{20\}\}\{10^\{20\}\}$$

Now, factor out $10^{20}$ from the numerator:

$$=\; \backslash frac\{10^\{20\}(10^2\; +\; 1)\}\{10^\{20\}\}$$

We can cancel out the $10^{20}$ term from the numerator and the denominator:

$$=\; 10^2\; +\; 1$$

Calculate the value:

$$=\; 100\; +\; 1$$

$$=\; 101$$

The value of the expression is 101.

The correct option is (c).

The standard form of a number is a way to express it using a single non-zero digit before the decimal point, multiplied by a power of 10. This form is particularly useful for very large or very small numbers.

To convert 12345 into standard form, we need to place the decimal point after the first digit (1) so that we have a number between 1 and 10.

Starting with 12345, we move the decimal point to the left:

• 1234.5 (1 move)
• 123.45 (2 moves)
• 12.345 (3 moves)
• 1.2345 (4 moves)

Since we moved the decimal point 4 places to the left, the power of 10 will be 4.

Therefore, the standard form of 12345 is $1.2345 \times 10^4$.

The correct option is (d).

We are given the equation $2^{1998} - 2^{1997} - 2^{1996} + 2^{1995} = K \cdot 2^{1995}$. To find the value of K, we need to simplify the left side of the equation and isolate K.

The lowest power of 2 in the expression is $2^{1995}$. We can factor out $2^{1995}$ from each term on the left side:

• $2^{1998} = 2^{1995} \times 2^3$
• $2^{1997} = 2^{1995} \times 2^2$
• $2^{1996} = 2^{1995} \times 2^1$
• $2^{1995} = 2^{1995} \times 2^0$ (or simply $2^{1995}$)

Substitute these back into the equation:

$$(2^\{1995\}\; \backslash times\; 2^3)\; -\; (2^\{1995\}\; \backslash times\; 2^2)\; -\; (2^\{1995\}\; \backslash times\; 2^1)\; +\; (2^\{1995\}\; \backslash times\; 1)\; =\; K\; \backslash cdot\; 2^\{1995\}$$

Factor out $2^{1995}$:

$$2^\{1995\}\; (2^3\; -\; 2^2\; -\; 2^1\; +\; 1)\; =\; K\; \backslash cdot\; 2^\{1995\}$$

Now, calculate the values inside the parenthesis:

• $2^3 = 8$
• $2^2 = 4$
• $2^1 = 2$

So, the expression inside the parenthesis is:

8 - 4 - 2 + 1

Perform the subtraction and addition:

= 4 - 2 + 1

= 2 + 1 = 3 Now the equation is:

2^{1995} (3) = K \cdot 2^{1995}

To find K, we can divide both sides by $2^{1995}$:

K = \frac{3 \cdot 2^{1995}}{2^{1995}}

K = 3

The value of K is 3.

The correct option is (c).