Question

If $3^{x} \times 27^{x}=9^{x}+^{4},$ then what is $x$ equal to?
A) 4
B) 5
C) 6
D) 7

Answer 1

Indices are a convenient tool in mathematics to compactly denote the process of taking power or a root of a number. Taking power is simply a case of repeated multiplication of a number with itself while taking a root is just equivalent to taking a fractional power of the number. Therefore, it is essential to understand the concept as well as the rules of indices to be able to apply them later in important applications.

Complete step-by-step answer:
In the multiplication of exponents if the bases are the same then we need to add the exponents.
Consider the following:
$1.2^{3} \times 2^{2}=(2 \times 2 \times 2) \times(2 \times 2)=23+2=2^{5}$
Like that, in the above given example,
Given, $3^{x} \times 27^{x}=9^{x+4}$

$$\Rightarrow 3^{x} \times(3)^{3 x}=3^{x+4} \\\ \Rightarrow 3^{4 x}=3^{2} x+4 \\\ \ldots . . \text {Using } a^{m} \times a^{n}=a^{m+n}$$

As the bases are equal i. $e .3,$ their powers must be equal.

$$\text { Therefore, } 4 x=2 x+8 \\\ \Rightarrow 2 x=8 \\\ \Rightarrow x=4$$

Thus, the answer is option $\mathrm{A}: 4$

Additional Information
Negative powers: Consider this example:
$\dfrac{a^{2}}{a^{6}}+=+a^{2-6}+=+a^{-4}$
Also, we can show that:

$$\dfrac{a^{2}}{a^{6}}+=+\dfrac{1}{a^{4}}$$

So a negative power can be written as a fraction. In general:

$$X^{-m}+=+\dfrac{1}{x^{m}}$$

Power of zero: The second law of indices helps to understand why anything to the power of zero $\dfrac{+}{x^{3}} x^{3}+=+1$
Is equal to one. We know that anything divided by itself is equivalent to one. So $\dfrac{+}{x^{3}} x^{3}+=+x^{3-3}+=+x^{0}+=+1$
Also, we know that $\dfrac{+}{x^{3}} x^{3}+=+x^{0}+=+1$
Therefore, we have shown that
Fractional powers Both the numerator and denominator of a fractional power have meaning. The ground of the fraction stands for the type of root; for example, $x^{\dfrac{1}{3}}$ denotes a cube root $\sqrt[3]{x}$
The top line of the fractional power gives the usual power of the whole term.
For example:
$x^{\dfrac{2}{3}}+=+(\sqrt[3]{x})^{2}$
In general:
$x^{\dfrac{m}{n}}+=+(\sqrt[n]{x})^{m}$

Note: Laws of indices
The first law: multiplication: If the two terms have the same base (in this case $x$ ) and are to be multiplied together their indices are added.
The second law: division: If the two terms have the identical base (in this case $x$ ) and are to be divided their indices are subtracted.
The third law: brackets: If a term with power is itself increased to power then the powers are multiplied together.