Categories

## Problem on Equation | AMC-10A, 2007 | Problem 20

Try this beautiful problem from Algebra based on quadratic equation

## Problem on Equation – AMC-10A, 2007- Problem 20

Suppose that the number $a$ satisfies the equation $4 = a + a^{ – 1}$. What is the value of $a^{4} + a^{ – 4}$?

• $174$
• $194$
• $156$

### Key Concepts

Algebra

Linear equation

multiplication

Answer: $194$

AMC-10A (2007) Problem 20

Pre College Mathematics

## Try with Hints

Given that $4 = a + a^{ – 1}$. we have to find out the value $a^{4} + a^{ – 4}$

Squarring both sides of $a^{4} + a^{ – 4}$ …then opbtain…

can you finish the problem……..

$(a + a^{ – 1})^2=4^2$ $\Rightarrow (a^2 + a^{-2} +2)=16$ $\Rightarrow a^2 + a^{-2}=14$ and now squarring both side again………….

can you finish the problem……..

Squarring both sides of $a^2 + a^{-2}=14$ $\Rightarrow (a^2 + a^{-2})^2=(14)^2$ $\Rightarrow a^4 + a^{-4} +2=196$ $\Rightarrow a^4 + a^{-4}=194$

Categories

Categories

# What are we learning ?

[/et_pb_text][et_pb_text _builder_version=”4.1″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]Competency in Focus: Powers of Numbers This problem from American Mathematics contest (AMC 8, 2013) is based on basic  algebra and Powers of Numbers.[/et_pb_text][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

# Next understand the problem

[/et_pb_text][et_pb_text _builder_version=”4.1″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]If $3^p + 3^4 = 90$$2^r + 44 = 76$, and $5^3 + 6^s = 1421$, what is the product of $p$$r$, and $s$?[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.0″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”4.1″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px”][et_pb_accordion_item title=”Source of the problem” open=”on” _builder_version=”4.1″]American Mathematical Contest 2013, AMC 8 Problem 15[/et_pb_accordion_item][et_pb_accordion_item title=”Key Competency” _builder_version=”4.1″ open=”off”]Basic algebra and Powers of Numbers[/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.1″ open=”off”]4/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” _builder_version=”4.0.9″ open=”off”]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

[/et_pb_text][et_pb_tabs _builder_version=”4.1″][et_pb_tab title=”HINT 0″ _builder_version=”4.0.9″]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title=”HINT 1″ _builder_version=”4.1″]First, we’re going to solve for $p$Start with $3^p+3^4=90$. Then, change $3^4$ to $81$. Subtract $81$ from both sides to get $3^p=9$ .Now we can write 9 as $3^2$ .So, from here we can say that p=2.[/et_pb_tab][et_pb_tab title=”HINT 2″ _builder_version=”4.1″]Now, solve for $r$. Since $2^r+44=76$$2^r$ must equal $32$,  and 32 can be written as $2^5$ .So from here we have r=5.[/et_pb_tab][et_pb_tab title=”HINT 3″ _builder_version=”4.1″]Similarly now, solve for $s$$5^3+6^s=1421$ can be simplified to $125+6^s=1421$ which simplifies further to $6^s=1296$=$6^4$ , which gives s=4.[/et_pb_tab][et_pb_tab title=”HINT 4″ _builder_version=”4.1″]Lastly, $prs$ equals $2*5*4$ which equals $40$. So, the answer is 40.[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ min_height=”12px” custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]